,1. \V. Oihbs — Kijidlihriidii of n>fii'i><inivi>i(H Su/isfa )/<•,. t. 127 



tli:it (4">) shall lioM Inic tor all va^iati^•n^ in ihc slate of the system 

 ami ill tlie <]uaiititi('s ot' tlie various Hulistanees eomposiiitj it, even 

 tlioui;;li these v;iriati<>iis an- not consistent with the e(|nations of con- 

 dition (39), (40), (41), For, when it is not |>ossil)le \i, dn this, it 

 must be possil)le by a|i|)lyin«; (45) to variations in the HyKtcrn not 

 necessarily restricletl by the etjuations of condition (-M*), (40), (41) to 

 obtain conditions in re«:;ard to 7\ /\ .l/,, .1/,,, . . . M„, Home of 

 which will be inconsistent with others or with c(|nalions (4:i). 'I'liese 

 conditions we will repreBoiit by 



-1=0, 7?^ 0, etc., (JC.) 



.1, />, etc. beint; lineai- functions of 7', /', .1/,, .1/.,, . . .1/,. Then it 

 will be possible to deduce fioni these conditions a sinj^le condition of 

 the ft)rin 



n A + /i n + etc. ^0, (47) 



(K, fi, v\c. beini; positive constants, which cannot hold true consist- 

 ently with ecpuvtions (43). Hut it is evident from the form of (47) 

 tliat, like any of the eontlitions (40), it couhl have been obtained 

 directly from (4')) by applying this formula to a certain chanj.(e in 

 tlie system (|)erha])S not restrictcil by the ecpiations ot condition (30), 

 (40), (41)). Now as (47) cannot hold true consistently with eqs. (43), 

 it is evident, in the tirst place, that it cannot contain 7'or /*, there- 

 fore in the diange in the system just mentioneil (for which (45) 

 reduces to (47)) 



2^6>/ + ::: J>/f=^\ and 2: O/- -f 2:' />>/" = 0, 

 so that the equations of condition (39) aiid (40) are satisfied. Again, 

 for the same reason, the homogeneous function of the first degree of 

 JAj, J/o, . . . J/„ in (47) must be one of which the value is fixed by 

 eqs. (43). l)ut the value thus fixed can only be zero, as is evident 

 from the form of these equations. Therefore 



( >■ ()■/// , + :^' it/n , ) J/, + ( :i" 6)n , + ::i" Um^ ) j/, . . . 



+ ( ^" (h,}„ + :i lJni„) J/„ — (4 8) 



for any values of ^/^, M^ . . . J/„ which satisfy eqs. (43), and 

 theretore 



(I^'fJ///, + >Z>///i) 3j+ (:^'(J;«2 + ^"^'"2) 2j • • • 



+ ( >' d'i/,„ + >■ Jjjn„) 2„ = (49) 



for any numerical values of 2^^, Z2, . • . 3„ wdiich satisfy e«is, (3ft). 

 This equation (40) will therefore hold true, if for r of the letters 

 3,2.,.. 3„ we substitute their values in terms of the others 

 taken from eqs. (38), and therefore it will hold true when we use 



