On some Combinations of Boracic Acid with Oxide of Lead. 375 

 we may replace the values of B, and A, in (16.) by 



„ 2 f «7r Stir 1 



Jd; = — <TO,cos \-WqCOS [■""> 



' w [ * w ^ n j 



h . . (IV.) 



Af= — -^57, sin h^osm 1-.... ^ 



n \^ n ^ n (J 



Then if the particular values of/(^ —) be given, 



or 



tabulated, we may with very little trouble form tables of u^, 



u^y &c. ; t^i, ^25 &c. ; and from these last tables of w^, ttJg, &c. ; 



a-i, ^2J &c. Thus the labour of computation required by (15.) 



would be reduced to one half of the same in (16.), and to one 



fourth of it in (17.). 



We may reduce (14.) once in a similar manner, omitting 



2 

 the last terms in the values of B^ and A., which will be -f{2v) 



and respectively ; but this can be done only once. It ap- 

 pears, therefore, that (15.) is prefei'able to (14.) when we have 

 tables of the particular values. 



If we employ a smaller value of w, we shall have for certain 

 values of i 



iJc k k 



sin- =0, sin {p-i) ~ =0, sin (p + i) j =0; 



and we should find 



Bo + B„ +B2« + 



B^ + B„_i + B„+j + B2„_i +...».. 

 A.— A„_i + A„+<— Aj8„_< + 



in the place of Bq, B., and A. respectively. These might be 

 employed to determine some of the quantities B^ and A., when 

 a part of them has been found by the method before given. 

 Gunthwaite Hall, April 9, 1849. 



LV. On some Combinations ofBoracic Acid toith Oxide of Lead, 

 By Thornton J. Herapath, Esq.*. 



l^EUTBAL Borate of Lead, PbO+BO^, maybe obtained 



by digesting the heavy white precipitate which is formed 



when biborate of soda is added to a solution of any neutral 



salt of lead, for twelve or fourteen hours, in a strong solution 



* Read before the Bristol Philosophical and Literary Society, April 19, 

 1841, and now communicated by the Author. 



