Chief Justice Cockle on the Conversion of Integrals. 537 



it) would pass on with little obstruction. On the contrary, it is 

 proved by a beautiful set of experiments by Prof. Tyndall, that the 

 absorption of heat in passing through a gas varies within certain 

 limits as the density. Now this is the very law of absorption 

 which the authors of " Climate " assumed as their data ; and, 

 for the correctness of their mathematical results, it is immaterial 

 whether the assumption be that the absorption is due to the at- 

 mosphere, or to the aqueous vapour suspended in the atmosphere. 



I conclude, therefore, that Mr. Meech^s memoir will not give 

 us the annual terrestrial temperatures, and that I cannot believe 

 the results given in the admirable article " Climate " to be 

 " wholly erroneous " till better reasons are alleged. 



June 8, 1867. 



LXVII. On the Conversion of Integrals. By the Honourable 

 Chief Justice Cockle, F.R.S., President of the Queensland 

 Philosophical Society*. 



1- T JET _,, 



0, (1) 



and suppose that, by means of series or by any other means, 

 we can obtain a solution of (1) in the form 



y=\4)(x,v)dv, (2) 



where the integration signified is either definite or indefinite. 



2. The object is to supplant the integration, definite or inde- 

 finite, with respect to v by an indefinite integration with respect 

 to x. This can be done whenever we can find two functions of 

 x only, say e and f, such as to satisfy the relation 



t^t +» + *&*U • • • (3) 



in which, for brevity, cj> represents <f>(x, v) andjf^, v) may be 

 any function free from integration with respect to v which will 

 satisfy the relation. 



3. Taking the variables x and v as independent, and integra- 

 ting (3) with respect to v, we have 



J^& + et,<H(?-c)jVfo+/(*,t>) = X, . . 



(4) 



where X is a function of x only. But, by (1) and (2), this last 

 equation is equivalent to 



J+(?- e )2/ = X -^fe v)-f(x, v); ... ( 5 ) 

 * Communicated by the Author. 



