The Astronomer Royal's Reply to Professor Challis. 149 



ness of the process by which it has been obtained. With 

 this estimation, I beg leave" to call attention to the principle 

 hinted at in page 67, line 22, &c. It will be remembered that 

 <J> is a function which is obtained by solving a partial differ- 

 ential equation, and that (as always happens) it contains in its 

 expression another function^ whose form is arbitrary; i.e. 

 any form may be used for f, provided that the quantity which 

 is used as the subject of the function be that which is given by 

 the solution, viz. (r— at), and provided that the function chosen 

 be placed exactly in the place of f in the solution, and be 

 treated in no other way than that in which f is treated. Yet 

 Professor Challis says, "The function <p may thus contain 

 implicitly, as a factor, another function expressing the va- 

 riation of velocity at a given instant in passing from one 

 point to another in directions perpendicular to the motion, 

 but is not differentiated with respect to the variables of this 

 factor." And a little further on, having found an expression 



<p = J * ~ — L , he adds, " the factor cos 0, depending on 

 r 



the mode of disturbance, being included in the arbitrary 



function." That is to say, having found that a certain set 



of differential equations is satisfied by the expression 



<J> = *0 — + " _T — ~ — f' 9 Professor Challis assumes that 

 */ x^+y^ + z* 



we may use as a solution % •J\v x +ff + g ~ a > , and that 



ar + if + s 2 



in substituting it in the differential equations we have no oc- 

 casion to differentiate the factor ., * 5-.. Against 



4/ (or +#*■+%*) & 



this I must record my solemn protest. The function which 

 is to be put in the place of the arbitrary function is simply to 

 be put in its place, and to be modified in no other way what- 

 ever. If the solution in that form cannot be adapted to the 

 mode of disturbance or to the initial motion, it is a sign that 

 (among the infinity of solutions applying to fluids) one has 

 been chosen which is inapplicable to the conditions of the 

 problem, and another must therefore be tried. As to the omis- 

 sion of certain terms in the process of differentiation, I con- 

 fess that I am surprised. Any person so well acquainted as 

 Professor Challis is with the transformations of the equations 

 for fluids, must be aware that the introduction ofthe peculiar 

 function is a matter of convenience only ; that it does not 

 at all modify' the suppositions on which the fundamental equa- 



