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The heat capacity of a substance can be measured under adiabatic or non-adiabatic conditions. Modern experimental techniques allow measurements made under adiabatic conditions to yield values for the specific heat accurate to 0.1%. However, knowledge of the solid state at low temperatures is still in the prospecting stage and new information that is accurate only to 1% or 2% is valuable. The non-adiabatic method with its comparatively simple apparatus and ability to produce more rapid results is, therefore, a useful technique to employ. The measurements on solid hydrogen below 4°K., and irradiated polythene from 15°K. to room temperature, are good examples of the proper application of this method. The specific heat of solid hydrogen has been measured as a function of the ortho-para concentration from 0.2°K. to 4°K. There are two species of hydrogen molecule: ortho-hydrogen and para-hydrogen. The ortho-hydrogen molecule in its ground state is nine-fold degenerate, being three-fold degenerate with respect to both the molecular rotation and the nuclear spin. The specific heat anomaly due to the lifting of the ground rotational level of the ortho-hydrogen was found by Simon, Mendelssohn and Buhemann (1929) below 10°K. Hill (1956) repeated the measurements with improved experimental techniques, but was unable to make measurements below 2°K. using standard calorimetric practices, due to the spontaneous heat of conversion of the ortho- into para-hydrogen. The total anomalous entropy per mole of ortho-hydrogen above 2°K. was only 0.8 E.U. of the expected 2.18 E.U. (R Log 3). In consequence, the experiments reported here were undertaken to extend the temperature range of the measurements down to 0.2°K. This was accomplished by a novel design of calorimeter. The calorimeter was made with two compartments. The first contained liquid helium and had a large diameter pumping tube attached to it; the second contained a powdered paramagnetic salt, iron alum, and the hydrogen specimen, which was condensed between the grains of the salt. By rapid pumping on the liquid helium it was possible to reduce the temperature of the calorimeter to 1.3°K., after which a demagnetization of the iron alum reduced the calorimeter and the sample temperature to 0.2°K. A time-temperature curve was taken until the calorimeter was above 2°K., as the rate of warm-up was too rapid for conventional specific heat points to be made. As it could be shown that the heating due to the ortho-para conversion was sensibly temperature independent, it was possible to calculate the specific heat of the sample from the slope of the heating curve and a knowledge of the heating rate. The heating rate was found from conventional specific heat measurements made above 2°K. The results showed that at ortho-concentrations greater than 60%, a modified λ-anomaly existed in the temperature range 1°K. to 2°K.The form, height, and the temperature of the maximum varied with concentration. For 73% ortho-hydrogen, the peak is at 1.60°K. and 5 calories.degree-1.mole-1 high, while with a sample of 63% ortho-hydrogen, a small peak, 1 calories.degree-1.mole-1 high, appears with a maximum as 1.12°K. At ortho-concentrations below 60%, there is no λ-anomaly. The specific heat becomes appreciable at about 0.2°K., rises steeply until a temperature of about 0.6°K. has been reached, after which it rises slowly to a maximum between 1°K. and 2°K., before falling steadily towards the triple point. Whether a λ-peak occurs or not, the total entropy of the anomaly is the same within experimental error. The arithmetical average of the entropy for the different runs is 2.14 E.U., which is in excellent agreement with 2.18 E.U. expected theoretically from a lifting of three-fold degenerate state. The existence of a λ-point is consistent with the changes in the nuclear resonance spectrum reported by Hatton and Rollin (1949) and Reif and Purcell (1953) for normal hydrogen, and Sugawara et al (1955) for other ortho-hydrogen concentrations. These experiments only indicate whether or not transitions occur among the substates. They would have obtained the same low temperature resonance pattern, indicating now transitions among states, if the molecules were frozen into arbitrary substates. It is only the existence of the specific heat anomaly of the correct entropy, that indicates that an ordered state is obtained at very low temperatures. These experiments also show that there is no indication of the entropy of mixing of the ortho- and para-molecules. Further, the rate of change of ortho-concentration was found from the libration of heat due to the ortho-para conversion. If o is the ortho-concentration expressed in percent and t is the time in hours, then do⁄dt = 1.79×10-4o2. The constant is in good agreement with Cremer and Polanyi's value of 1.9×10-4. The specific heat of irradiated Polythene was measured from 15°K. to room temperature with standard calorimetric methods. This sample was the most heavily irradiated out of a set, the remainder of which was measured by Smith (1955). The specific heat per gram was smaller over the whole temperature range than the other polythenes. A theoretical calculation of the specific heat of normal polythene was made, which seemed to indicate that the rapid rise in specific heat of the polythene above 180°K. was due to molecular vibration at natural CH2 characteristic frequences, and not to a disorder change in the crystallites of the polythene, as had been assumed by previous workers. The specific heat of Potassium Bromide was measured from 4°K. to room temperature. A comparison with the measurements of Clusius, Goldman and Perlick (1949) indicated that their values between 40°K. and 80°K. were probably too high. The change in the Debye θ at low temperatures could be explained qualitatively with reference to Blackman's (1933, 1937) calculations.