14-4 The Astronomer Royal's "Reply to Professor Challis 



tion, and that (with due care in integration) they do therefore 

 contain the remaining equations of the same order. Thus 



Professor Challis's equation (2.\ or ^-(--^\ ^~(^\ 

 1 ■ a z\ dx J dx \d% / 



,.«, . . . . <P / d P \ 



gives, on differentiation with • respect to y, -j , ( ~r~ ) 



— —, — r- (—,— ) ; hut this same result may be obtained by 

 dxdy \dz J J J 



differentiating my equations (1.) and (3.), or — 7 — ( — - } — ) 



d /dP\ . d /dP\ d /dP\ J_ - 



= T7 (^> «** -j- j^j-J = ^ ^ j. Professor 



Challis will find, on examination, that I have used all which 

 were required. 



But Professor Challis states that these six equations " are 

 exactly verified by the kind of motion which Mr. Airy con- 

 siders to be impossible " (page 64, line 29), and then proceeds 

 to show (page 61$) that Poisson's law of motion satisfies these 

 equations. What particular obscurity in my expressions, or 

 what omission of my explanatory words in Professor Challis's 

 reading of them, can have led to such a misinterpretation of 

 my meaning as this sentence implies, I cannot imagine. My 

 words (page 322, line 15) were, " If, in order to support Pro- 

 fessor Challis's expression for the movement of the particles 

 to or from a centre, we suppose, &c. &c." I did not object 

 to Poisson's expression, or to any other involving cos as a 

 multiplier; but I remarked (page 328, line 17) that "the 

 onus of proving that the three equations are consistent rests 

 with the supporter, &c." ; in other words, that the possibility 

 is not to be assumed without proof, and that each case must 

 rest on its own proof. That I did not intend to object to 

 every expression involving cos #as a multiplier in the formula 

 for the resolved part of the velocity in the direction of radius, 

 will be sufficiently evident from this circumstance; that the 

 simplest of all kinds of motion, namely, that of a uniform 

 current of air, in which all the particles move with the same 

 velocity in parallel lines, is included in this case. My ob- 

 jection was specially to " Professor C/iallzs's expression." 



Professor Challis, in the proof that Poisson's assumed mo- 

 tion is possible, has preferred referring to the function <p, where 



-= — = u = velocity in the direction of x; -~- = v = velo- 

 dx J dy 



