556 MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 



we seek the values of &t, £J, £J, we are met by the difficulty that the sum of 

 the three expressions is zero, and that virtually we have only two equations 

 whereby to determine three unknown quantities. We are thus forced to bring in 

 the condition 



A^ + A^ + A^ = «' > 



by help of which we obtain the rather complex values 



At = 3 e + g e J 2 cos (t \f ^ \ j , 



A* = 3 « + 3 « ( - cos (t */-A - v /3 . sin (t \f ^\ j , 

 "{-cos^y^- J3 . sin (* v^) } . 



1 i 1 -i 



53. To return from this digression to the subject of quaternary functions, we 

 observe that 



sus t = jT] t + [JJ t , cat t = [7] t + [a] t 



cost = [o]t - \T\t , sin t - [TJ t - Q , 



wherefore 



[Tj £ = j sus / + cos t | 



1 f -> 



[TJ t = „ < cat ^ f sin t \ 



\±\t = nl sus /f — cos t \ 



[JJ ^ = = j cat < — sin M 



Now, all recurring functions of the fourth order are expressed by the general 



formula 



A[T|* + BjJJ* + G\J]t + D0; 



in which A, B, C, D are any numerical coefficients ; and, consequently, they may 

 also be expressed by 



a sus t + b cos t + c cat t + d sin t , 



so that the theory of quaternary recurring functions becomes a compound of the 

 well-known doctrines of Trigonometry with the analogous and complementary 

 doctrines of the catenarian, or, as some may prefer to call them, the hyperbolic 

 functions. It would be easy to multiply formulas connected with these functions, 

 many of them interesting, on account of their relations to other researches ; but 

 as my present object is only to indicate the general features of the inquiry, I 

 shall leave these, and proceed to apply the quaternary functions to the solution 

 of a problem in Mechanics, which has resisted all the powers of the integral 

 calculus. 



