OF ARTS AND SCIENCES. 39 



The last term in equation (9) disappears when the planet passes 

 at equal distances north and south, and in general is inappreciable ex- 

 cept when the error of i^osition zero is large, or when the planet 

 passes at very unequal distances north or south of the centre, which 

 in practice need never occur. 



Equation (9) consequently permits the determination of the 

 diameter by observations on one limb only. As has been remarked, 

 observations on the preceding limb before inferior conjunction, and on 

 the following limb after it, may be expected to eliminate errors pe- 

 culiar to the elongation. 



It should be remarked that the quantity cot p cot \ p becomes 

 unity for p = 60°, and for larger angles rapidly increases. In general 

 the advantageous application of the method requires the use of lines 

 at greater position angles than 60°. 



In what precedes it has been assumed that the line corresponding to 

 the time t^ bisects the angle formed by the others, and also passes 

 through their intersection. In ruling the plates for the observations 

 of the diameter during the transit of Yenus, these conditions may 

 possibly not have been exactly fulfilled ; since, as they did not affect 

 the observations then contemplated. Professor Rogers did not especially 

 attend to those points in the preparation of the plates. Any such 

 errors may, however, be eliminated. Thus, if we put a = the dis- 

 tance between the transit line on the plate from an imaginary line 

 parallel to it passing through the point of intersection of the inclined 

 lines, and A^ the inclination to a line bisecting the angle of the in- 

 clined lines, equation (3) becomes, 



^» = '«+ IT^ [± i^ + ^^ i^P + ^i) +-] 



and equation (9), 



Z> = =F V- cos 8 cot jo cot 1 JO Ft -f t' — 2 a — 2 (AS — A'S) At 



Since Ai changes sign by turning the plate 180° in position angle, 

 and a changes sign by turning the other side of the plate toward the 

 eye these sources of error may be determined or eliminated by arran- 

 ging the observations with appropriate reversals. 



To exemplify partially the use of this method, I avail myself of 

 some observations of the following limb of Venus, on various days 

 succeeding the recent transit, by Professor Rogers and myself. As 



