VOL. XC.] PHILOSOPHICAL TRANSACTIONS. QQQ 



this climate. It is to be noticed, that the nights must be very clear; the moon 

 absent ; no twilight ; no haziness ; no violent wind ; and no sudden change of 

 temperature ; then also, short intervals for filling up broken sweeps will occasion 

 delays ; and, under all these circumstances, it appears that a year which will 

 afford 90, or at most 100 hours, is to be called very productive. 



In the equator, with my 20-feet telescope, I have swept over zones of 2°, with a 

 power of 157 ; but an allowance of 10 minutes in polar distance must be made, for 

 lapping the sweeps over each other where they join. As the breadth of the zones 

 may be increased towards the poles, the northern hemisphere may be swept in about 

 40 zones: to these we must add 19 southern zones ; then, 59 zones, which, on 

 account of the sweeps lapping over each other about 5 ra of time in right ascension, 

 we must reckon of 25 hours each, will give 1475 hours. And, allowing 100 hours 

 per year, we find that, with the 20-feet telescope, the heavens may be swept in 

 about 14 years and -§-. Now the time of sweeping, with different magnifying 

 powers, will be as the squares of the powers ; and, putting p and t for the power 

 and time in the 20- feet telescope, and p = 1000 for the power in the 40, we shall 



have jb 2 : t : : p 2 : -j = 5Q840. Then, making the same allowance of 100 hours 



per year, it appears that it will require not less than 598 years, to look with the 

 40-feet reflector, charged with the above-mentioned power, only one single mo- 

 ment into each part of space ; and even then, so much of the southern hemisphere 

 will remain unexplored, as will take up 213 years more to examine. 



V. A Second Appendix to the Improved Solution of a Problem in Physical Astro- 

 nomy, inserted in the Philos. Trans, for \7Q&, containing some further Remarks, 

 and Improved Formula for computing the Co-efficients a and b ; by which the 

 Arithmetical Work is considerably Shortened and Facilitated. By the Rev. John 

 Hellins, B. D., F. R. S., &c. p. 86. 



It was shown, in art. 9, of the first appendix, how the common logarithm of the 



fraction *• ~ CC , when c is expressed in numbers, may be taken out from the 



best log. tables. Yet that method of obtaining the value of «, easy as it is, requires, 

 first, a search in the table for the angle of which c is the sine, and generally a pro- 

 portion for the fractional parts of a second ; then, a division of the degrees, minutes, 

 and seconds contained in that angle, by 2 ; and, thirdly, another search for the 

 logarithmic tangent of half the angle, and another proportion to find the fractional 

 parts of a second. Mr. H. was therefore desirous of finding some easier and shorter 

 method of performing the whole business, without the use of any trigonometrical 

 tables, in which time is required, not only in searching for logarithms, but also in 

 making proportions for the fractional parts of a second ; and, after some considera- 

 tion, he discovered that which is here explained. This method then, together with 

 some further observations made for facilitating and abridging the work of computing 



