DERIVATIVES ASSOCIATED WITH LINEAR DIFFERENTIAL EQUATIONS. 409 

 whence, after logarithmic differentiation, 



and, therefore, the foregoing quantity is, as stated, an invariant of index unity. 

 Taking it in an integral form, we have as a new invariant 



e*.M,i = /*e,,el-xe A M , -. . . . . . . . (vii.) 



of index X -f- p. -\- 1 ; from the similarity of its form with the Jacobian of two binary 

 qualities, it will be called the Jacobian of 8 A and 8,. As we have already seen, 

 quadriderivatives of composite functions are composite functions when Jacobians are 

 retained. 



36. The number of Jacobians which we need to retain as independent of one 

 another is diminished by the two following results : 



(A) If either of two functions be composite, their Jacobian is composite. For, 

 taking 



*, = e^e,, (x + P = <r), 



we have 



<*>. , = M ex - (x + P ) 8,8,8,, 



= fi f , M, 1 "f- *M*A, * 1. 



a composite function. 



(B) Of the \(n 2)(n 3) Jacobians 8^, derived from the primin variants 

 only n 3 are algebraically independent ; for between the three which are 

 derivable from any three priminvariants 0^, 6,,, 8, we evidently have a 

 relation 



Hence the independent Jacobians, as well as the quadriderivative functions 8 ffi are 

 all of the second degree in the coefficients of the differential equation, and may 

 therefore be called quadrinvariants. And from the foregoing it follows that the 

 aggregate of " proper," i.e., non-composite, quadrinvariants is composed of two sets, 

 which are 



(i) the n 2 quadriderivative functions 8,,, given by (vi.) ; 



(ii) the (n 2) (n 3) Jacobians 8^, given by (vii.), of which only n 3 

 are algebraically independent of one another. 



37. The simplest example that occurs is in the case of the quartic equation of 

 which the canonical form is 



MDCCCLXXXVIII. A. 3 G 



