30 



RELATIONS PERTAINING SIMPLY 



[BoOK 1. 



previous injunction, that, if the variations of the angles v and y are conceived to 

 be expressed, not in parts of the radius, but in seconds, either all the terms con 

 taining d v, d y&amp;gt;, must be divided by 206264.8, or all the remaining terms must be 

 multiplied by this number. 



29. 



Since the auxiliary quantities (f, E, M, employed in the ellipse obtain 

 imaginary values in the hyperbola, it will not be out of place to investigate their 

 connection with the real quantities of which we have made use : we add therefore 

 the principal relations, in which we denote by i the imaginary quantity y/ 1. 



l 



sin cp = e = - 



COS lp 



tan (45 } 9) = 



= * tan 



tan (p -- | cotan (45 i 9) i tan (45 i 9) = -- : 



cos f/3 = i tan y 



(f = 90 -f- f log (siii 9 + 1 cos 9) = 90 log tan (45 



tan i E= i tan i F = *&amp;gt;fll) 



+l 



i 



= I cotan 



tan 



:= z cotan F, 



or 



or 



cotan E= % cotan J^ 



r-. 



SWJ! 



-\-l 



2 M 



or 



- =: log (cos E -f- * sin E] = log -, 

 J? = Hog w =: i log (45 -f- 



The logarithms in these formulas are hyperbolic. 



