^sin>'-a) smja'-©') /J__ Ja 



P *'-* sin («"-<*'/ tf sin(a"-a')'\r' 3 R'V 



determining the Orbits of Comets. 131 



neglecting the higher terms dependent on the interval of time, 

 the formula will assume this shape : 



t* - V 



(*"-a,y~V sin (a" -a') 



A form quite similar will be obtained by combining the first 

 with the four last of the equations (13). Eliminating from 

 them p' we have 



» — [ r ' r "] tari 8 ' cos (* — ©') — tan S cos Q ' — ©') 

 = [rr'J ' tan I" cos (a'— 0*) — tan S'cos («"— 0') e 



__ [r'r"] RtanS'cos(0— ©')— [rr'^R'tanV+irr 1 ] R"tan S'cos(©"-©') 

 [r r'] (tan a" cos (*' — 0') - tan V cos (a" — 0')) 



As we before introduced the Y, so we may here introduce 

 the X in reference to the direction ©', and by the same pro- 

 cess we shall obtain 



„_ t" — t' tan 8' cos (a — .0') — tan 8 cos (a' — ©') 

 ° ' " t' — t ' tan 8" cos (a' — ©') — tan 5' cos (a" — 0^) ^ 



i 



XT' 



R' tan 8' / 1 1 \ 



tan 8" cos (a' - ©') - tan 8' cos (a" -e')\r T3 ~ W 3 ) * 



Both equations are correct to quantities of the first order 

 inclusive. Their combination gives, by eliminating the term 



which has the factor 1-^ — JL j, the equation (16). 



The proceeding, which in the above-mentioned case has hi- 

 therto appeared to me the most convenient, and which has in 

 practice proved to be so as much as could be expected, (it 

 being by almost two orders less accurate than the original 

 formula for M,) consists, therefore, in the following: If the 

 value of M appears to be too indeterminate, on account of the 

 smallness of its numerator and denominator, one may choose 

 in place of it either 



M -*""*' sin & ~ «> or 

 M -*<-;- sin (*"-«<)' 



M - *" ~ I tan 8' cos (a -©') - tan 8 cos (a' - ©') 

 M ** f —t ' tan 8" cos (a' - 0') - tan 8' cos (a"-©') ' 



according as the differences of geocentric longitude or latitude 

 are the more considerable, and consequently allow less influ- 

 ence to the possible errors of observation. The error hereby 

 introduced is of the first order of the intervals of time. The 

 calculation is then carried on to the completion of the trials, 

 by which approximate values of r and r" become known en- 

 tirely by formulae (I) and (II). The value of r is then found 



S2 



