Royal Institution. 303 



anything else but a rational function of {s) and {t)j and thus the 

 theorem is completely demonstrated. 



It may for a moment be objected^ that we have been dealing 

 only with a particular form x'^-\-Qmoc'^y^-\-y^, instead of the 

 general form ax'^ + 4ibx^y + ^cx^y^ + Mxy^ + ei/^ ; but the latter is 

 always reducible to the former by means of a definite linear sub- 

 stitution j and if we call the modulus of the substitution [i. e. the 

 square of the determinant formed by the coefficients of substitu- 

 tion] M, to every general invariant I^ of the qth. degree, to the 

 latter corresponds a partial form (I ) of invariant to the former, 

 such that 



and consequently, since every (I) is a rational function of {s) 

 and [t), so must every I be the same of s and t; unless, indeed, 



it were possible to have I^/ = — 7.(F),^' being different from and 



greater than q: but if this were the case, since ^q=-T7^(Iq) ^ 



power of M, the modulus would necessarily be an invariant ; 

 but in passing from x'^ + y"^ + 6mx'^y^ to x'^-j-y'^-\-6y{7n)x'^y'^, 

 1 + 3m becomes the modulus, which we know is not an invariant ; 

 hence the proposition is completely established for the case of 

 the biquadratic function {x, «/)'**. 



[To be continued.] 



XL VIII. Proceedings of Learned Societies, 



ROYAL INSTITUTION OF GREAT BRITAIN. 



Feb, 11, /^N the influence of Material Aggregation upon the ma- 

 1853. ^^ nifestations of Force, by John Tyndall, Esq., Ph.D. 

 There are no two words with which we are more familiar than 

 matter said force. The system of the universe embraces two things, — 

 an object acted upon, and an agent by which it is acted upon ; — the 

 object we call matter, and the agent we call force. Matter, in cer- 

 tain aspects, may be regarded as the vehicle of force ; thus the 

 luminiferous aether is the vehicle or medium by which the pulsations 

 of the sun are transmitted to our organs of vision. Or to take a 

 plainer case ; if we set a number of billiard balls in a row and impart 

 a shock to one end of the series, in the direction of its length, we 



* I have made a tacit assumption throughout the foregoing demonstra- 

 tion (which is, however, capable of an easy proof), viz. that if any fractional 

 function of the coefficients of any form be invariantive, the numerator and 

 denominator must be separately invariants. 



