372 Mr Glaisher, On the functions [Feb. 1, 



same apparatus which had been used for the cyanide. The result 

 corresponded with anticipation. The thermometers in the bath 

 and in the flask all rose together until the bath was at 147°, the 

 mixture at 110° and the vapour at 75°. When the bath reached 154° 

 the mixture fluctuated from 108° to 115° and the vapour was at 

 77°, but on raising the temperature of the bath to 155° the ther- 

 mometer in the mixture fell suddenly to 90° and that in the vapour 

 to 65°. It has long been recognised that in the continuous process, 

 whereby a small quantity of sulphuric acid gradually converts a 

 large quantity of alcohol into ether and water, the continuity of 

 the reaction must be due to differences of temperature in different 

 parts of the liquid. The directions for making ether given in 

 handbooks of chemistry direct that the mixture is to be maintained 

 at about 140°. This will certainly imply a much higher temperature 

 for the retort and for the layer of liquid in contact with it. Our 

 experiment seems to indicate that the reaction between alcohol 

 and ethyl-sulphuric acid by which ether is produced begins at 155°, 

 or at least begins to occur quickly at that temperature. With the 

 bath at 154° the temperature of the vapour was nearly that of 

 boiling alcohol, and it dropped when the ether began to come ; 

 but of course it would never drop to the boiling point of ether 

 because as much water as ether is formed in the reaction and both 

 are vapourised together. Moreover Thomsen's observations shew 

 that this reaction is attended with a sensible evolution of heat. 



(2) On the functions inverse to the second elliptic integral By 

 J. W. L. Glaisher, M.A. 



Consider the function inverse to ez x, where ez x is the Jacobian 

 form of the second elliptic integral given by the equation 



rx 

 ezx=l dn 2 xdx. 



Jo 



Let this function be denoted by ea x, so that, if 

 drfxdx = u, then x = ez~*u = ea u, 



f 



Jo 



the letter "a" in the functional sign ea suggesting the word 

 amplitude. 



It follows that, if 



and if 



<"* V(l - ftV) 



o V(i-*0 



/, 



I A (<£) d<f> = u, then <f) = am ea u, 

 Jo 



dx = u, then <£ = sin am ea u. 



