IN THE THEORY OF DIFFERENTIAL EQUATIONS. 



527 



\f/ b = 0, is the condition under which •fyjAa + ^/ bx db = is satisfied whenever \f/ a da + \js b db = 

 is satisfied. Consequently, the criterion of detection, when satisfied, completes all that is 

 wanted to make a and b compensatory to the second order. 



10. I defer for a moment some details peculiar to the biordinal equation, and I give a 

 sketch of the method of treating triordinals. Let y = \j/ (m, a, b, c), combined with y = \j/ x , 

 y" = ^x»> give a = A (x, y, y, y"), b = B (x, &c), c = C (x, &c.) ; and let y'"= x (x, y, y, y") 

 be the deduced triordinal. As before, the identities dA:A y „={y" -y^dx, &c, show that 

 AL, = co , &c. severally give singular solutions, except when satisfied by a relation void of y" : 

 that is, they give all the singular biordinals. Substitute A for a, &c, in y = \J/, y = \p„ 

 y"= -j/ XI , which makes them identities, and gives A y „ from the first three of the four equations 

 following : 



^ax A V + ^6, B »" + fcx Cy = °. 



^™ A r + ^« B v + ^« Cy = »• 



* 



t „4,, + ^ bxxx B y „ + >// c „ x C^, - Xy-> 



A,„ = 



T . 



T w " ^JvM'c.r* + fax^bxx^c + ^axx^b^cx 

 ~ ^c^bx^axx ~ i'cxfbni'*- ^cxWVrV- 



The general property of the singular biordinals is T = 0, and cases involving 

 y ! / b x l / cx~ ^c^b* ~ °» & c * must ^ e considered apart, and are related to singular primordinals of 

 the singular biordinals. Make the substitutions in y'"= \|/ JOT , which brings ^,„ to identity 

 with y, and we have the fourth of the equations. In T ui , write one more x in each suffix, 

 advance each other letter one place in the cycle abca, and change the sign : thus •^^ bx ^\> cxz 

 becomes — ^^cxW'Wr' & c - '• ca ^ tne resu ^ T. Repeat this process twice, striking off" xs 

 when they reach four in number in any suffix : call the results T and T It . Multiply the four 

 equations by T, T f T \ f T uj> and add, which gives T lt + T iu y^,m 0. Now it will be found 

 that T = — {T )»; whence \ y .,= (logT^ x . Hence may be established, relatively to 

 v ,„ conclusions similar to those of the preceding cases : and M . Cauchy's theorem may be 

 extended. 



It thus appears that the generalizations of form which a complete theory would require 

 may be easily obtained by those who have studied the new and as yet isolated method of 

 determinants* ; that is, by all who have taken full advantage of existing facilities. 



The implicit forms (p (x, y, a) = 0, (p (x, y, a, b) = 0, &c, may easily be substituted for 

 y = \|/ (x, a), &c. We find 



* I should rarely venture to resist the term introduced by 

 investigators into their own results, except by passive refusal to 

 follow them : but the present case will allow even a remon- 

 strance. Very good leason ought to be shown before the 

 participle determinant is separated from the verb determine : 

 and till such reason is manifest, I, for one, feel unwilling to 

 abandon the right to say that co-ordinates are determinants of a 

 point, and coefficients of an equation determinants of the roots. 

 That some instances of such separation are too well established 

 to be shaken, is unfortunately true: the evils which they 

 create, in thinking and in teaching, are warning enough against 

 the introduction of more. The rule of invention, hitherto, has 

 Vol. IX. Part IV. 



been to let the name express the most striking distinctive 

 character of the thing signified. Determination is no distinc- 

 tive property of the so-called determinants: all functions deter- 

 mine. But eliminating power is their distinctive property, and 

 elimination is the process in which they were first thought 

 worthy of a name. Until those who have a right to speak with 

 authority propose a distinctive appellation, I intend to call 

 them eliminants. But, though objecting to the word determin- 

 ant, 1 feel the highest admiration of the system ; it is difficult 

 to say how much will hereafter be held due to those who have 

 laid its foundations. 



68 



