Mr. J. J. Sylvester on Aronhold^s Invariants. 299 



- If elementary books of geometry were, like those of more 

 modern sciences, accommodated to the character of the age, it 

 could not, I presume, have happened that any one should have 

 felt puzzled by such theorems as the above. The general rela- 

 tions of the subjects considered would then be better understood. 

 The theorem now discussed is only a modified instance of one 

 more general, viz. that in any triangle the smaller angle has the 

 larger bisector. Both may be deductions from the following 

 position, viz, that of all straight lines passing through the same 

 point in the bisector of an angle, and limited by its constituents, 

 that which has the smaller inclination to the bisector is the 

 longer, and those that are equally inclined to the bisector are 

 equal, and conversely, by direct demonstration, those which are 

 equal are equally inclined. 



From these truths we may derive the modified case relating 

 to the isosceles triangle, either directly or indirectly ; the indirect 

 mode being, as is the case generally when it is properly employed, 

 the easiest and the most elegant. 



[To be continued.] 



XL VII. A Proof that all the Invariants^ to a cubic Ternary Form 

 are Rational Functions of Aronhold's Invariants and of a connate 

 theorem for biquadratic Binary Forms. By J. J. Sylvester, 

 F.R.S.f 



ALTHOUGH contrary to the order of exposition indicated in 

 the title to this paper, I shall, as the simpler case, begin 

 with establishing the theorem for a biquadratic form, say F in 

 x^y. Let 



F = «a;4 + 4Z>a^2/ + 6ca? V + ^(^^^ + ^2/^ 

 s—ae—4ibd+Sc^ 

 t=ace — ad^'-(^—b^e-\-2bcd, 



s and / are the two well-known invariants of F. I propose to 

 prove that there can exist no other invariants to F except such 

 as are explicit rational functions of s and t. 



Let F, by means of the substitution of fx-{-gy iov oc, and 



* Communicated by the Author. 



t A Constant in analysis is any quantity which in its own nature, or by 

 the explicit conditions to which it is subjected, is incapable of change. An 

 Invariant is an expression apparently liable to change, but which, owing to 

 certain compensations in the modifying tendencies impressed upon it, 

 remains as a whole unaltered. The former may be compared to a fixed 

 point or system in mechanics ; the latter to a point or system free to move, 

 but kept at rest under the combined operation of contending forces. 



X2 



