^9^ On the Trains of Clods an J TFatcbes. 



on the axle of the contrafe" wheel; and the third portion will then be the contfate wheel, 

 with the pinion adluated by it, the crown wheel, and the palats : the firft portion determines 

 the time that a watch fhal! go at one winding up, as has been obferved before; the fecond 

 portion determines the mutual velocities of (he minute and fecond hands; and the third por- 

 tion regulates the number of beats per fecond : hence any one of thefe portions may be con- 

 flrufted varioufly, without afFciSting the calculations of the two others. 



If a watch be required to have 6 fpirals on tha fufce, and to go juft 30 hours, we fliall 



have for the firft portion 6x — , 6 X — , 6 X — , and 6x — : for the fecond portion wc 



9 10 II 12 



48 41: A.!. 48 6d 42 42 60 60 48 48 60 60 

 may take --. x -^, or -^^ x ^ , -, X V,'or 1- -^ -~, — y. \, ot2^ -x. -~, — x 

 6 6 6 667 7 06 o 8 66 



C4 C4 60 60 60 60 60 60 -40 4Q 60 60 c6 56 60 

 ^,or i5 X r-, -^ X — ,or— x — , — X -,or^ ^ — , - X ^g-^orir ^ "' 

 9 9 66 10 10 67 7 7 77 8 o 7 



63 60 60 63 J , n, 11- 60 ^4 ^4 6° .-, •/• t 



— X ,or— X — ; andj laftly, as already given, ,,- •X — ,or — x —• likewife, for 

 7997 .,8000 



. . . 64x15x2 56XICX2 48x15x2 r I.- u 



the third portion, we may put— ^ — ^ -, = , and , any one of which 



formula, divided by 60, the feconds in a minute, will give 4 for the number of beats per 

 fecond. 



Here then we have gotTou'r varieties of the firft portion, nine of "the fecond, without in- 

 verting the order, and three of the third, which, by the rule for finding* compcfitions, will 

 produce 4x9x3=108 different forms of contraftion, to anfwer the fame purpofe, without 

 making any alteration in the number of fpirals on the fufee, or in the dial-work ; and where 

 no number out of all the wheels exceeds 64, or is lefs than 42. 



The allowance to be made in corredling the feconds gained by the beats of any watch 

 muft be , J^o of the whole, for every minute in the daily error in the rate of goingj plus or 

 minus^ accordingly as the rate is too flow or two faft : therefore, in the fecond note {Nj, 

 page 5c, where, for an error of five minutes, it is faid that the allowance will be " }efs than 

 3^0" of the whole, k ought to have been " upwards of jio:" for t+'+75= j-^s is a greater 

 portion of unity than jioj though inadvertently it might be taken as conftituting a fmaller^,, 



It may probably appear too fornlal to notice here two typographical errors in the defcrijjp, 



tion of the numbers propofed for a new Wajch in my laft paper, where " 2 pivots" ought tp 



have been *' 2 palats," (or " pallets," or " palettes," for different authors fpell this woijfi 



djflerently) ; and 6 fpirals on the " barrel," 6 fpirals on the " fufee;" but it is abfolutely ne- 



cefTary to adva-t to another paflage therein, wiiich, partly from my own ifjaccuracy, which. 



you have already noticed, and partly from the printer's omiflion of the figns + and — , .as^it 



now ftands» is completely unintelligible. At the afterifk in page 53, let the fentence be U/>) 



dcrftood to be thus : as 23 h. 56 m, 4,9:98 f. (the length of a fidereal rotation of the earth)^ 



: 'j-J-ELi'j^ m 1 .iji 'jiii /,' J ' , •, 



* Vide Hutton's CourTe of Mathematics, vol. i. p. 134. "i 



..flus 



