180 Mr. A. Smith on the Calculatmi of the Distance 

 3. The equation to the cone of the earth's shadow is 



{a?2+y+^^-R^}{s^-R'} = {«^+2'j/+c^-R2}2 . (1.) 



Also, since 



a:=f) sin ?cosa «=SsinZcosA 



y=psin5sina J=SsinZsinA 



2;=fCos^+R' f=ScosZ, 



it follows that 



^^2_|.j^2^^2=p2^2RpC0S^+R^ . . . (2.) 



ax 4- i?/ + c^ = Sp { sin ?sin Z cos ( A — «) + cos ?cos Z } + SR cos Z 



= (by spherical trigonometry) Sp cos <p + SR cos Z. . (3.) 



It is the introduction of (p at this part of the process which 

 so much simplifies the result. 



Substituting in equation (1.) the values of x^ + y'^ ■\- z'^ and 



o^ ax-\-hy + c?^i given by equations (2.) and (3.), and putting gr 



■p 

 for ^T^, q being a small quantity, the average value of which is 



about 'OOie, we have 

 f)2{sin«(p-y2}4.2Rp{(l-g2)cos?-cos^(cosZ~y)} 



-R2{cosZ-g}2=0 



} (4.) 



In this formula sin (p may be considered as always greater 

 than (7, or <^ greater than 16'; since, if smaller, it would be 

 impossible to make any approximation to the distance of the 

 shooting star. This equation, therefore, will have two roots ; 

 one positive, the other negative. The positive root is of course 

 the only admissible solution. Solving the equation and ma- 

 king R = l, we obtain 



-n/{^ 



-g^)cos^- cosf(cosZ-<7) '\^ {cosZ-qf 



^ (1—9^)008 ^— cos <p (cos Z — g) . - V 



sin2<p-<72 ^^'> 



The calculation ofp from this formula will be facilitated by 

 the use of a subsidiary angle 4', such that 



CP,tifr- (l-g')cOS^-COSf(cOsZ-g) , ^ ^ ^gj 



-/ sin^ <p -- g^ ( COS Z — g') 

 we then have 



cosZ— » ^ ^ ,^v 



vsin^(p — ^-^ ^ 



4. In almost all cases a sufficiently approximate result will 



