SECT. 2.] TO POSITION IN SPACE. 59 



Jl a= 195 47 40&quot;.25 ft = -337 40&quot;.02 



1= 65528.98 log cos ft 9.9991289 



logr 0.3259877 log cos I Q ... 9.9832852n 



log cos ft 9.9991289 9.9824141 



log/ 0.3251166 log cos u 9.9824141. 



The calculation by means of formulas III., VII. would be as follows : - 



log sin u .... 9.4454714w log tan i* 9.0604259 



log sin 9.3557570 log tan 8.8020995 



log sin ft . &quot;7 .. . 8.8012284^ Iogcos M 9.9824141 n 

 ft= 337 40&quot;.02 log sin (u I -f Q, ) . 7.8449395 



u l + Q = 024 3&quot;.34 



IQ, = 195 47 40.25. 



52. 



Eegarding i and u as variable quantities, the differentiation of equation III., 

 article 50. gives 



cotan ft d/5 =: cotan idi-\- cotan wdw, 

 or 



XII. d^ r=sin (X Q ) d -(- sin z cos (X 2 ) dw. 



In the same manner, by differentiation of equation I. we get 



XIII. d(Jl Q) = tan/3cos(Jt Q)dt + ^d. 

 Finally, from the differentiation of equation XI. comes 



Ar = cos ft dr rsin/fd/3, 

 or 



XIV. dr = cos/?dr r sin ft sin (X Q ) d r sin ft sin i cos (X &) du. 



In this last equation, either the parts that contain dz and du are to be divided by 

 206265&quot;, or the remaining ones are to be multiplied by this number, if the 

 changes of i and u are supposed to be expressed in minutes and seconds. 



