338 Carnot on the Theory of 



diftance we may fuppofe to intervene betwixt the lines RSl 

 and MP. 



Now, it is eafy to fee that this laft equation is fufceptiblft 

 of the following form : 



/ TP y \ /rr^ yMZ + aR Z — xRZ \ 



\ y a- x) + V y (a-x) . (2a-zx-MZ)J 



an 



* As this equation appears to be erroneous, I (hall examine it 

 throughout ; putting, for brevity, TP — s, TT — s, MZ = x, 

 RZ =y. The equation immediately preceding, in this notation, is 

 s + / zy +y s I 2y + j 



T .— 



y 2a <-~ Zx — x y y 2,2 — 2x — x 



which being made — o, is [ — + — ] — ( : ) =0. 



v y y ' \ia — zx — x> 



v* s y s y 



Now, (by article 9) s = , or — • = — — , or — =oi 



a — x y a — x y a — x 



and thiswillbethefirft memberof the new equation, which is to contain 

 what our author calls " affigned" quantities only, as the fecond member 

 is to contain none but " auxiliary" quantities. The firft part of the 



fecond member is obvioufly — , an auxiliary quantity • and, in order 



to render the fecond part, — ( • ) alfo ivbolly auxilf- 



V 2<2 — zx — x> J 



ary, we multiply both the numerator and denominator by a — x (or 



yy 

 by its equal —-, by art. 9). This multiplication by a—x, gives 



the latter part of the numerator ay— xy; and thus far the author's 

 r.ew equation is right. 



But the auxiliaiy, equivalent to the affigned quantity %y . (a — x) t 

 cannot be what he makes it; for, as we have jull feen, a — x = 



Vv 2v^v 



~-, and therefore it fhould be zy . (a — x) = . 



x x 



Now the author has zy . (a—x) = y'x, and confequently, if bis 

 Tt/ult be right, we fhould have, 



— : — = yx, and y = — r. 

 x Zy 



■d . „ • \ V * , ax—xi 



But (by art, 9, again) — - — = — , and y = , 



a — x y y 



Therefore = — , and a— x == — ; 



y zy 2 



But this being plainly impoffible, I conclude that zy . (a—x) — 

 ■z-J, or zy z y : x ( = zy* . (RZ : MZ) in our author's notation) 



and 



