Prof. Cayley on Differential Equations of the Second Order, 197 



The velocity v of any film as represented by equation (12), 

 and the efflux Q R as represented by equation (17), are given in 

 terms of the velocity v of the central filament, and can only be 

 determined by these formulae when that velocity is known. The 

 experiments of M. Darcy having made it known in certain cases, 

 they serve to verify the theory as it regards the relation of the 

 efflux per unit of time to the velocity of the central filament. 

 If that velocity could be represented in terms of the head of 

 water and the diameter and length and degree of roughness of 

 the pipe, then the formula by which this was represented being 

 substituted for v in equation (12) would complete the inves- 

 tigation. I propose to make it the subject of a future com- 

 munication. 



XXIII. On a supposed new Integration of Differential Equations 

 of the Second Order. By Professor Cayley, F.R.S.* 



I CANNOT assent to the views taken by Professor Challis in 

 his paper in the July Number, " On the Application of a 

 new Integration of Differential Equations of the Second Order 

 to some unsolved Problems in the Calculus of Variations." 



In any problem of the calculus of variations, where there are 

 two variables x y y, the condition for a maximum or minimum is 

 obtained in the form 



A(fy-po>)=0; 



and if the problem involves no relation between So? and 8y, Pro- 

 fessor Challis says that "we have with equal reason A = and 

 Ap = 0; }:> and he goes on to argue that "it cannot be antece- 

 dently affirmed that these are identical equations ;" and further, 

 that " it is necessary to take account of results deducible from 

 them either separately or conjointly." 



I object to this statement ; it seems to me that in order that 

 A(8y—p&ai) may vanish, the only condition is A = 0; we are 

 not concerned with the equation A/?=0, as such, at all. But in 

 certain cases it happens that p is a multiplier of the differential 

 equation A = 0; viz. by writing this equation under the form 

 Ap = we have an equation integrable per se, and which by its 

 integration gives the integral of the differential equation A=0. 



Professor Challis, taking the view that the two equations are 

 distinct from each other, considers (Problem II.) the following 

 question : — " Required the minimum surface generated by the 

 revolution of a line joining two given points in a plane passing 



* Communicated by the Author. 



