548 MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 



all quaternary functions, 02 2 — 202 . 02 + 02 2 = constant, 

 as also 02* — 202 . 02 + 0|2 2 = constant. 



47. By substracting equation 10 from equation 9 we obtain 



1= 0I 2 + 2Q« . 02 + 0* 2 - 0* 2 - 2 02 . 0* - 02 2 , or 



1 = (0* + 0*} 2 -10' + 0O 2 ; (11) 



now 02 + 02 is evidently the function sus 2, while 02 + 02 is cat 2, wherefore 

 the above equation amounts to the property of catenarian functions that sus 2 2 — 

 cat 2 2 = 1. 



48. By adding together the same equations we have, on the other hand, 



1= 2 2 - 2 2 .0*+ 0* 2 + 0* 2 - 2 2 . 2 + 2 2 , or 

 1 = {0* - 0<} 2 + {0* - 0«}' • (12) 



The differences 02 — 02 and 02 — 02 will be at once recognised as cos 2 and 

 sin 2 respectively ; and I shall at once give these titles, without, however, in the 

 meantime attaching any signification to them, that is to say, for shortness' sake 



we shall put 



1 - * = cos t ; t - t = sin t (13) 



and then equation (12) becomes 



1 = cos t 2 + sin t 2 (14) 



Taking the derivatives of equations (13) we have 



t — t = . cos t = — sin t 1 



*-* (15) 



t — t = x sin t = + cos t ) 



and thus these functions cos 2, sin 2 are recurring functions of the Fourth Order, 

 the order of derivation being 



cos t , — sin t , — cos t , + sin t ; + cos t , etc. 



49. If we measure distances along the line of abscissae, corresponding to 

 successive values of the primary variable 2, and, if we set up ordinates propor- 

 tional to the values of the four functions, we shall obtain four curves characteristic 

 of these functions. For the case 2=0, the curve must pass through the point 

 A (fig. 11), situated at the distance 0A equal to the linear unit. 



Since the first derivative of 02 is there zero, the curve must at A be 

 parallel to the line of abscissae ; and, since its second derivative 02 is also zero, 

 the radius of curvature at A must be infinite, that is to say, the curve must be 

 quite flat at A. Moreover, the third derivative 02 being also zero, and its fourth 

 derivative, viz., itself, being positive, A must correspond to a minimum ordinate, 

 and the curve must rise on either side of A ; also the curve must be symmetri- 

 cally placed on either side of the axis 0A. 



The curve passes through the origin 0, and since the derivative of 02, viz., 



