232 Notices respecting New Books. 



Higher Algebra;" the work before us, however, discusses the pro 

 perties of binary forms only, i. *. the properties of homogeneoT*- 

 f unctions of two variables so and y, and does not enter upon th } 

 more general question of functions of three or more variables 

 After four preliminary chapters on the Symmetrical Functions 01 

 the Roots of Equations, on Resultants (i. e. Eliminants, the result 

 of eliminating x between two equations), on Discriminants, and 

 on Canonical Forms, the author discusses the properties of Inva- 

 riants and Covariants, which are the main subject of his work. If 

 we take the form 



a x 2 -\-2a^i/-\-ay, (1) 



and subject it to linear transformation by substituting pX. -f- 7 Y 

 for x, and p'X + ^'Y for y, it becomes 



A X 2 + 2A l XY + A 2 Y 2 ; 

 we shall now find that 



aa-a^ (pq'-pmw-o \ 



so that the function a a 2 —a* of the coefficients of (1) undergoes 

 no change in the transformation beyond being multiplied by the 

 square of the determinant of the transformation, viz. (pq'—pq)"; 

 consequently the function a a 2 —a* of the coefficients of (1) \i 

 called an invariant of that form. If we take cubic, quartic, or 

 other forms, functions of their coefficients can be found which have 

 the same property of invariancy. 



When forms of higher degrees than the third are considered, i 

 is found that they have several invariants, which in terms of the 

 coefficients of the form are long expressions. In fact the invariant 

 (cp) of the rth. degree of a binary form of the nth. degree can be ex- 

 pressed as the sum of all products of powers of the coefficients such 

 as 



where 



. m Q -f 1 . m l + . , . + 7im n = ^ nr, 

 and 



^o + ^i . .. + m n = r : 



the indices and suffixes must be taken to satisfy these equations ii 

 every possible way ; and then the constants (C) can be determinec 

 by the relation 



da x da 2 da n 



It is plain from this statement that, if r and n are moderately 

 large numbers, the invariant will consist of a large number o 

 terms. Thus in the case of the quintic form (n = 5) the invarian 

 of the fourth degree (V=4) consists of the sum of twelve terms 

 while the invariant of the 18th degree (n=5, r = 18) consists of 84£ 

 terms. 



It may be inferred from this that the subject branches rapidb 

 out into a variety of complicated developments. Thus we see tha 



