THE EQUILIBRIUM OF BODIES IN CONTACT. 487 



Hence, by Lagrange's theorem, we liave from equation (60), neg- 

 lecting powers of a above the first, 



tan - = tan — 

 2 2 



I {(cosS-o-cos e)+ (sin^|cot|^+cos^|tan|-°] (4'„-e)-:Jcosycot4' jtan ■ 



2 



COS" — tan — -sin^ - cot — 

 2 2 2 2 



u 



Where ^„ is taken to represent the value of •i' when a = 0* so 

 that by equations (60) and (61), 



Now, 



*'"''2 =M^^ V^-sin^eJsec^ 



I (6^)- 



(cos % - COS O) tan -^ - 2 sin i(% + 9) sin i (4'„ - O) sin^ — 

 ___^ ^__ = 2 _ _ a ■ « *« 



cos^|tan|-sin^|cot| cos^|sin^|-sin=|cos^| ~ ''" ^' 



cos-|tan|-°+sin^|cot|-° tan=| + tan=| 

 cos= I tan I - sin^l cot |" tan^'|-tan^| ' 



COSH cot ^„ i(]-tan-^|-»)(l-tan^ 



2 



cos^ -2 tan -» - sin^ - cot -» tan= l2 _ tan= ^ 



• By Lagrange's Theorem, if y = z + a^j/, then, neglecting powers of « above the first, 

 and representing by J'y any function of t/. 



Let y^F^, <Py = F,-*, f,j = F,^; ...z = F%„ <pz = F,%, Jz = F.f^. 

 F.i' and F,* be any otlier f 



F,^ = F,^^.F,%{'L^')a. 



If therefore /'S' = FS'„ + cF.i' and F^f be any other function of ^, then, neglecting 

 powers of a above the first, 



