62 CALIFORNIA ACADEMY OF SCIENCES. [Proc. p Ser. 



It is now easy to describe the structure of the group. 

 The infinitesimal transformations U ik W> f for p=i,2, .. 

 m — i generate a self-conjugate sub-group G\ of G. For if 

 we form ( U {k W) , U^ a ) ) f, whenever it is not zero, it is found 

 to be expressed in terms of U^ ^' f. Compounding 



therefore all of the infinitesimal transformations of G with 

 those of G\, only operations of G\ result. Therefore G\ is 

 a self-conjugate sub-group of G. It has (m — i)w 2 para- 

 meters. Similarly taking U ik W) f for p = 2, 3 . . . m — 1, we 

 find that these transformations generate a self-conjugate 

 sub-group G 2 , with (m — i)ir parameters, of G\, and so on. 

 We are thus finally led to an ;/ 2 parameter group G m _ 1 

 which is self-conjugate sub-group of a 2 « 2 parameter 

 group G m _ 2 , and which is generated by the infinitesimal 

 transformations ^7 ik (I " ~ l) f- The finite equations of the group 

 are 



Vi = yi , vi =yi ,   Vi {m -* ] =y i {m - 2] , 



Vf m ~ 1] = a*y x + a i2 y 2 -j- . . + a iQ y u -f y/— 1 '. 

 (t = 1, 2, . . »). 



§ 4. Linear Differential Equations. 

 If the transformation group of a linear differential equa- 

 te Dn of the m n th order is the group G, it is easily seen 

 that the integration of that equation can be reduced to 

 the successive integration of the m equations of the n th 

 order, out of which the equation # of the m n th order is 

 compounded, and quadratures. The same is true for the 

 more general group which is found if in equations (1) the 

 coefficient of y^ in the expression for -^(A) is arbitrary 

 instead of being equal to a ik . The group G however has a 

 special significance owing to the fact that if rj 1 , . . rj n and 

 a ik are considered as functions of a variable x, the group of 

 linear substitutions 



n 



Vi = 2 a ik y k 



k=l 



extended m — 1 times has the form of the group G. 



University of California, 

 December 17, 1898. 



