MR EDWARD SANG ON FUNCTIONS WITH RECURRING DERIVATIVES. 531 



These inverse functions are only new forms for well-known logarithmic ex- 

 pressions, thus 



—i 



BUS X 



—1 



log {x + V (x 2 — 1)} ; cat x = log {x •+ */ O 2 + 1)} 



sec x = log 



1 + y (1 - x 2 ) 

 x 



- 1 , 1 + V (1 + x 2 



cse « = log - 



tan x = | log 



1 + x 



Y^x 



cot a; =: i log 



X 

 X + 1 



# 



1' 



but they serve to give unity of structure to those alternate integrals which are 

 possible or impossible in circular functions, according as x is greater or less 

 than unit. 



19. The analogy of these compound catenarian or hyperbolic functions to 

 the trigonometrical lines, is clearly shown by their geometrical representatives — 

 thus, having measured OA, OB each equal to unit from the two sides of a right 

 angle, let fig. 1, OC be laid off equal to some value of sus t, and draw the ordinate 



Fig. I. 



CD equal to the corresponding value of cat t, then D is a point in the equilatera 

 hyperbola of which OA, OB are the two semi-axes. 



If we draw the radius- vector OD, and suppose D to be carried to a small 

 distance along the curve, the surface of the sector AOD will be augmented by a 

 quantity which, in all curves, is represented by J {OC . <5CD — CD . $OC} , where- 

 fore, in the present instance, the increment of the sector AOD is ^ (sus t . sus t 

 —cat t . cat t] dt, which, since sus f — cat f = 1, becomes ^ dt, therefore twice the 

 sector AOD represents the primary variable t. Through A and B draw two per- 



VOL. XXIV. PART III. 7 E 



