302 Prof. Challis on a new Method of solving some 

 and we have 





dx 



i 



== 2^ 7r |^Erf(«V^) cos 2ab + *^^ C e^ cos 2ab db 



^ — 1 ec smzaodo > t . {2d) 



a result which can be verified by forming the differential equation 



?/ denoting the integral, which, on integrating and determining 

 the constants by the considerations that the result must be in- 

 dependent of the sign of b, and that when b=0 it must equal 



2 s/7re a2c 



ErfWc), 



gives the same expression for y. 



The result (25) may be written in another form for 



Erf (a k/c + ~) = C f (* + T^dx 



— e c\ e~ x [ cos— 7-— 2'Sin-— )dsc. 



so that 



bi\ „ '/ bi\ i, b lr™ „«, 2^ 



Erf(Wc + — }+ErfU /V /c--9 1 = 2^ e -* 2 cos~-^, 

 / hi \ / hi \ b2 r™ Ihr 



Erf fa^/c- ^J-Erf(^Vc+-~) = 2ie* I r^siii^^ 



XXXIII. O/z « new Method of solving some Problems in the Cal- 

 culus of Variations, in reply to Professor Cayley. By the Rev. 

 Professor Challis, M.A., F.R.S* 



IN an article in the Number of the Philosophical Magazine 

 for September, Professor Cayley has expressed his dissent 

 from the new method of solving certain problems in the Calcu- 

 lus of Variations which is contained in my communication to 

 the Number for July. On carefully considering all that he has 



* Communicated by the Author. 



