150 The Astronomer Royal's Reply to Professor Challis. 



tions are obtained ; that these fundamental equations are the 

 four given by Professor Challis at the bottom of page 64 ; 

 that they are established on consideration of the complete 

 and absolute change in the state of pressure, density, and 

 velocity, of the particles of the fluid, in proceeding from a 

 point where the coordinates have one value to another point 

 where they have a different value ; and that it is absolutely 

 impossible that there can be, in forming them, any limitation 

 as to the functions whose change is or is not to be taken 

 into account. I am quite certain that the new doctrine of 

 omitting certain functions in the differentiation cannot stand 

 a moment, when examined with reference to the original un- 

 derstanding on which the first equations have been obtained. 



Perhaps the error of this principle may be made more ob- 

 vious by considering a simpler case. Suppose we consider the 

 motion of parallel plane waves through air. It is well known 

 that the disturbance of a particle in the direction of x may 

 be expressed by X = cos « . <p (u t — x cos « — y cos /3 — z cos y). 

 Suppose that there is a wall in the air, whose equation is 

 x = 0. The value of X must therefore vanish when x = 0. 

 But it does not vanish then. What must be done to make it 

 vanish ? A follower of Professor Challis would say, " Mul- 

 tiply it by a function which will vanish when x = 0, as for 

 instance a x, and omit the variation of this factor in all differ- 

 entiations." (This mode of escaping from the difficulty is 

 not at all more forced than that adopted by Professor Challis 

 in the passages to which I have adverted.) But I am sure that 

 Professor Challis himself would say, " Not so ; the process 

 proposed is illegitimate; the failure of the expression (in its 

 susceptibility of adaptation to the circumstances at the surface 

 of the wall) is to be remedied, not by multiplying it by a func- 

 tion which vanishes there, but by altering the whole formula, 

 so as to preserve the condition of satisfying the original dif- 

 ferential equations, and also to satisfy the new condition re- 

 lating to the wall." And thus he would obtain the formula 

 X = cos a . <J> (v t — x cos a. —y cos /3 — z cos y) 

 — cosa . $ (vt + xcos ct—y cos /3 — g cos y). 



I will only, in conclusion, express my regret at finding my- 

 self compelled to place myself so distinctly in opposition to 

 my excellent friend Professor Challis. Had the writer been one 

 of lower character or in a less influential position, or had the 

 subject been one familiar to a greater number of mathema- 

 ticians, I should have let it pass. But in observing the publi- 

 cation of principles the most dangerous to the honesty of ma- 

 thematics (if I may use such an expression) that I have ever 



