101] NON-INTEGRABLE CONSTRAINTS. 315 



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222a) M^~ t {x ( + a ( sin # cos ^ #' cos # sin ^ </)} + I = 6, 



b) M ^{y 1 -f a( sin # sin ^ #' 4- cos # cos ^ i//)} + p = 0, 



c) ^- {(If a 2 + ^4.) #' Ma sin # (V cos ^ + ?/' sin ^)} 



4- (Jf a 2 A) ^' 2 sin # cos # 



+ Csin#-^'(gp' + ^'cos'0') + Jf a cos # #'(#' cos ^ 4-0;' sin ^) 



4- Jtfa sin ^ ^' ( a' sin ^ 4- y f cos ^) = -M#a cos ^, : 



d) ~-{(Ma 2 cos 2 & + J.sin 2 #)^/ 4- cos # (^ ' + $ ' cos ) 



4- Jfacos^l x 1 sin^ 4- y f cos^)} 



4- Ma sin # #' ( x' sin ^ 4- y' cos ^) , 



Ma cos & ^ ( #' cos ^ f/' sin ^r) = 0, 



>o 



e) C(q)' 4- ^'cosd-) - 



We must observe that if we had taken account of the equations 217) 

 in the expression 221) for the kinetic energy, before differentiating, 

 we should have obtained quite different equations. Having performed 

 the differentiations, however, and introduced in the equations all the 

 reactions belonging to the different coordinates, we may now take 

 account of the equations of constraint, thus introducing, in effect, 

 the statement of the equilibration of some of the reactions, and 

 causing some of the terms to drop out. 



Now introducing the values of #', y\ from 217) in 222), and 

 eliminating A, \L from 222 a, b, e), we obtain 



223) Ma 2 1 sin ^ -j- (sin ^ <p' -}- sin # cos ^ &' + cos # sin ^ -^ f ) 



4- cos if; --T. (cos ^ cp ( sin # sin iff #' 4- cos # cos 



4- C^(9/4-cos#.^) = (), 



which, on performing the differentiations, and cancelling some of the 

 terms, becomes 



224) (Ma 2 -\- C)j t (y'+cos& <//) - Ma 2 sin & - #V = 0. - 

 Making corresponding simplifications in 222 d) it becomes 



225) ~l(Ma 2 + C)cos&(cp'+cos& </>') 4- ^sin 2 # 

 4- Jf a 2 sin >' = 0. 



