ON THE THEORY OF NUMBERS. 295 



of an automorphic is also an automorphic. (Tlie positive powers of a trans- 

 formation are, of course, the transformations wliicli arise from compounding 

 it continually with itself; its negative powers are the positive powers of its 

 inverse. See Mr. Cayley's Memoir on the Theory of Matrices, Phil. Trans, 

 vol. cxlviii. p. 17.) Hence, if a form have a single automorphic, of which no 

 two powers are identical, it will have an infinite number of automorphics. 

 The importance of automorphic transformations in the solution of the 

 problems of equivalence and transformation will be apparent from the 

 following considerations. If/" and/^ be two equivalent forms, \h\a. given 

 transformation of/j into/o, | ^i | and ja^j the general formulas representing 

 all the automorphics of/j and /^ respectively, all the transformations of 

 /j into /j will be represented by either of the formulae | a^ | X | A | or 

 I A I X I a^ |. And again, if/^ contain/^i a»d if we represent by | A^ |, ] A^ |» • • ■ • 

 certain particular transformations oi f^ into/,, obtained by compounding 

 each transformation (C), which gives a form <p equivalent to/i with some 

 one transformation of f into/,, then all the transformations of/ into/ will 

 be comprised in a finite number of formulae of the type 



|A,|xla,|, \h,\x\a,\, \h^\x\a^\, , 



I a, I still denoting indefinitely any automorphic of/. Or, if we employ the 

 second method of the preceding article, the same transformations will be 

 represented by 



\a,\x\hn, kJx|A,'|, la/lxl/i/l 



where | Kj ] is any automorphic of/, and | h^' |, | hj |. | V I' ^""^ certain 



particular transformations of/ into/, obtained in a manner sufficiently in- 

 dicated by the method itself. It appears, therefore, that when we know all 

 the automorphics either of/ or/, we can deduce all the transformations 

 of / into /, from one of those transformations when / is equivalent to Jl, 

 and from a certain finite number of them when/ contains, but is not equi- 

 valent to,/. We may add, that when one transformation of two equivalent 

 forms, and the automorphics of either of them are known, those of the other 

 are known also, for we evidently have the equation 



\cc,\=\h\-'x\aAx\h\. 



82. Problem of the Hepresentation of Forms. — We give the enunciation of 

 one other general problem, which may be said to occupy a middle place 

 between the problems of the representation of numbers, and of the equi- 

 valence of forms. By using a defective substitution of the type 



^2=^2,1^ 1 + ^2,2* 2+ "2,n-^ »-r» 



^n^'^n.l* 1 + '^n.'i'^ 2+ ^^11,11-^^ n-ri 



a form/(a;i,a:2, ar„) may be changed into another/ (a/j, x\, ^\~^ 



of the same order but containing fewer indeterminates. The form/ is said 

 to be represented by/ ; and the representation is proper or improper accord- 

 ing as the determinants of the system do not, or do admit of any common 

 divisor besides unity. Our third general problem therefore is, "Given two 

 forms of the same order, of which the first contains more indeterminates 

 than the second, to find whether the second can be represented (properly or 

 improperly) by the first, and, if it can, to assign all the representations of 

 which it is susceptible." If the second form contain only one indeterminate 



