Mr. Herapath on the Integration of Differential Equations, 19 



tains of the Alps the temperature may be slightly increased by 

 a cause hitherto unnoticed. In the upper part of the secondary 

 formations covering the granite, there are beds of gypsum, 

 and this gypsum is anhydrous ; but when exposed to air and 

 moisture, it combines with water, and passes to the state of 

 common gypsum : and during this combination we may sup- 

 pose heat to be evolved, but the process must be extremely 

 slow, and the heat evolved must be totally inadequate to raise 

 the temperature of powerful streams to 1 26°. Saussure found 

 the temperature of the water in the lower part of the salt 

 mines of Bex, which are situated in the vicinity of gypsum, 

 to be four degrees of Reaumur higher than the mean tempe- 

 rature of the earth. It is not improbable, though Saussure 

 was not aware of the circumstance, that this small increase of 

 temperature in the mines of Bex might be partly owing to the 

 combination of water with gypsum : however, an increase of 

 temperature it is well known, is observed in deep mines, far 

 removed from the gypsum formation. 



In reply to what I have advanced respecting the thermal 

 waters in the Pennine Alps, it may be said that few thermal 

 springs have been yet discovered in the northern range of the 

 Alps which form the Bernese Oberland ; but the difference in 

 the geological structure of the two ranges will, I conceive, be 

 sufficient to explain why hot springs are more rare in the lat- 

 ter, than in the southern range. Most of the highest moun- 

 tains in the Bernese Alps are covered with secondary strata ; 

 and the valleys are chiefly excavated in these strata, or in enor- 

 mous beds of sandstone and conglomerate, that form a thick 

 intervening mass between the surface and the primary rocks, 

 sufficient to obstruct the rise of thermal waters ; for it has be- 

 fore been stated, that all the thermal waters in the Pennine 

 Alps issue from the primary rocks, or near their junction with 

 the lowest calcareous strata, 



Hampstead, Dec. 8, 1827. Robert Bake well. 



IV. On the Integration of Linear Differential Equations having 

 Constant Coefficients and last Term any Function of the In- 

 determinate Quantity x. By John Herapath, Esq, 



[Concluded from vol. ii. p. 425.] 



Y^E shall now proceed to a few exemplifications of the 

 J * formulae. 

 Ex, 1. Suppose n = 2 then (17) becomes 



y = e r "{c 1+ fe( r - r ' )x {c+fXe- r * 



D 2 the 



