and therefore 

 Hence — 



260 NATURE [Jan. 16, 1890 



Now, since y(<|)) = w'(sec i^)"\ we have, as before 



/M = l{J»\ + m tan 7. 



/" F(<^)(/(/) ^ j yi^)^?!/) = tan 7 + j\(a - /8)2(sec 7)2r/('_^'\ + w + i tan 7I ; 

 and in the particular case where /= 2, and tn = 2, we have — 



I = ta„.^ + A(. - «)Vc7)f 2(i|)_^ + 3 >a" r] 



= .a„{, + ,V.-«=[.(-fi.)^ + 3.an.]|. 

 Hence the anfjle which the chord of the arc makes with the axis of jc is — 



7 + -,V(« - ^^^[^(^X + 3 tan 7I = 7, suppose. 

 Muhiplying by the value of X found above, we have— 



^=l(;^)C'^2-^r_3)(cos7)«-i|tan7 - ^S(«-i8)2 | (^)£4'^l(tan 7)2-4(sec 7)'] + tan 7[^^'^^^M^(sec 7)' " 6(sec 



-w-i « + 3j/ / ' 



^(.r^2)(^2 -^,i.)(cos 7)«-^ {tan 7- ^\(« - fir{{^)[4>-I^2isecyr - 4^7=7] + tan 7 [^7^-2 ;^s ^'^^ ^^^ 



- H-i « + 3j I }• 

 Considering - — ?— , _?_ - -J — , and o - ^ to be small quantities of the first order, the above expressions for 



—- - ^ , X, Y, and T are true to the fourth order. 



q" p» 



The quantity { ~ \ which occurs as a factor in some of the terms of the third order may be put under a very convenient 



form in the following manner. 

 We have, by Taylor's theorem. 



In this make co = ^(o - /8) and - J(o - ;3) successively ; therefore 



or 



Y 



and 





Hence we have to the first order of small quantities — 



~ ' p — (J _ (du\ 



a~- fi ~ V^/o' 

 and 



, , , ^ Up + 1) = «(,■> 



and therefore 



V«/#/o (/ + ;;')(« - )8) 



Making this substitution forf ~\ the expressions for X, Y, and T become— 

 \ua<p/o 



^ = .&(«- 2) (^^ -/'^y^°^^^'"4' " '^•f+f(«-^)tan7-'^'(«-3)n« + 4(sec7y^ - «T3]}; 



Y=_^— .^ (-1- - -'L-Vcos7)"-i|tan7-|.-^::i2(a-)8)[7^(sec7)2-"^^i]-.T\(a-/8)2tan7[^e^^T5(seC7)--«-l « + 3]l; 



'' = M^){-^-^ - ^y^^yy-'b- ^^(«-3)tan7 - ^(a-^)n«-Ti(sec7)^-.7+l]}; 



and these values are still true to the fourth order, considering -^^^ and o— ;3 to be small quantities of the first order as before. 



P + 1 

 The angle which the chord of the arc makes with the axis of x becomes, in like manner — 



7 = 7 + ^l^'lia-P) + i(«-8)2tan 7, 

 which is true to the third order. 



