IN THE THEORY OF DIFFERENTIAL EQUATIONS. 517 



compensative : thus <p x dx + <j> y dy + <p a da = is merely the condition of permanence of the 

 equation <p (x, y, a) = const., if the expressed letters be all which enter the form (p. When, 

 having (f> (x, y, a, b) = 0, we deduce <p x + <p y y = 0, which only expresses that w and y are 

 compensative, we usually imply da = 0, db = 0. It must be remembered that the word 

 compensation refers to form, not to value. 



3. If y be a function of x and constants which becomes infinite for a finite value of x, 

 then y':y = co , being (logy)' ; also y":y = oo , being (logy')'; and so on. These remain 

 true, even if y, &c. become infinite by the reduction of y = y (x, a, &c.) to as = m, exclusive 

 of y. That is, when a finite change in x makes an infinite change in y, it makes 

 an infinite change in y':y, y":y, he. This, as well as y':y = co when y = 0, may be made 

 visible from the subtangent of the curve represented, and demonstrably obvious both from 

 the fluxional principle, and from the equation <f> [x + h) = (px + <p' (x + Oh) . h. 



If, for a finite or infinite value of x, either or both P and Q become infinite, then P:Q 

 and P X :Q X are equal: that is, both nothing, both finite and equal, or both infinite. (Would 

 equatable be a good word to distinguish this set of alternatives from that in which the zeros or 

 infinites must have the ratio of equality ?) This is well known when both are infinite. Let 

 P be finite or zero, Q being infinite. If P x be also finite or zero, the theorem is obvious ; if 

 P x be infinite, P being finite, it is clear that, x being finite, the last increment of Q must be 

 described with an infinitely greater velocity than the last increment of P; so that P X :Q X = 0, 

 as well as P:Q. This fluxional step may be easily reduced into the language of limits. If 

 the critical value of x be infinite, then P cannot be finite or zero, unless P„ = ; whence, Q x 

 being anything except 0, P X :Q X = 0, as well as P:Q (this last because Q x (x = co ) not 

 being = 0, Q must be infinite). There only remains the case in which Q x = (x = co), P 

 being finite and therefore P x = 0, and Q being infinite. Let x be taken very great, and let it 

 then receive the increment Ax. The hypotheses show that AP:AQ diminishes without 

 limit, when Ax increases without limit: but this lies among the values of P X :Q X as found in 

 passing from x to x + Ax ; if then P X :Q X tend towards any final limit, that limit must be 

 zero. In all that precedes, we suppose that some tendency is permanently established at last : 

 in all other cases, the above theorem depends upon the interpretation of cos co and sin co . 



In my last paper I made use of the theorem that any relation between x and y under 

 which u is finite and u x infinite, gives either u y = co or x = const. This is equally true when 

 u is infinite; for the differential coefficient becomes infinite whenever the function becomes 

 infinite on a finite hypothesis involving the variable of differentiation. And the theorem may 

 be generalized thus : If v and w be any two of the subject-letters of a function u, then any 

 supposition which makes u v infinite either makes u w infinite, or is itself independent of w. 

 Hence, if a relation containing w make u x = co , we see that 



u x dx + u y dy + u z d% + ... = and u xl dx + u xy dy + u xz d% + ... = 



are the same equations: for they are 



dx + -= dy + — dz + ... = and dx + —dy + — dz + ... = 0. 



Ux v* u TX u XI 



