94 Prof. Challis's Researches in the 



less powers of this quantity above the first be neglected. In 

 this case the equation becomes 



and at the same time the equation (B.) becomes 



It is, however, important to remark, that the equation (A'.) 

 is satisfied without any restriction of the value of f the more 

 exactly in proportion as the value of /approaches more nearly 

 to unity. We may hence infer that the equation (B.) is 

 strictly true for points on and immediately contiguous to the 

 axis of the ray, but not more generally. Thus, with the ex- 

 ception just named, J have not hitherto succeeded in obtaining 

 equations applicable to ray-vibrations beyond the first order of 

 approximation, which probably is all that will be required in 

 application. In equation (3.) of my communication to the 

 Philosophical Magazine for last November (p. 364), the terms 



involvmg -7-^, -—■, and m, which arose trom relymg too im- 

 plicitly on the hypothesis by which udx -irvdy + 'wdz vm^ made 

 integrable, must be rejected. The equations (1.) and (2.) in 

 p. 363 hold good. It may also be stated, that the reasoning 

 in the December Number, by which it was shown that along 

 an axis of raj'-vibrations a given state of density and velocity 

 may be propagated uniformly and without alteration, remains 

 untouched. 



Having stated all the modifications which, according to the 

 present investigation, the previous results require, I beg to add 

 a few general remarks. It is well known that on first apply- 

 ing the integrals of partial differential equations to physical 

 questions, a dispute arose between Euler and D.'Alernbert as 

 to the arbitrariness of the forms of the functions. The dis- 

 cussion issued in the establishment of the principle of discon- 

 tinuity by Lagrange. Yet where two mathematicians of so 

 great eminence differ, it may generally be concluded that a 

 share of truth is on each side. The singular contradictions 

 which I have pointed out as resulting from the suppositions 

 of plane-waves and spherical waves, must, I think, raise the 

 question whether, as D'Alembert appears to have supposed, 

 forms of solution are not discoverable which indicate motions 

 independent of the particular disturbance, and whether these 

 must not be discovered before proceeding to consider cases 



