344 iPHILOSOPHlCAL TRANSACTIONS. [aNNO I/O/. 



>i— 1 



Here it is to be observed that, if — — ^ should be an odd number, the sign 



of the root when found must be changed to the contrary sign. 



Let there be proposed the equation 5y — lOy^ -j- \Qy^ = 6 ; here n = 5, and 



a = 6; then the root is y = 4.^6+ V35-|- , "^ Or, because 6+ v^35 = 



y(' O + /y/ 35 



11.916, its log. will be I.0761304, and its 5th part 0.2152561, the arith- 

 metical complement of which is 9.7347439: hence the numbers answering to 

 these logarithms are 1.6415 and O.609I, the half sum of which is 1.1253 = y. 



But if it happen that a be less than unity, then the second form of the root 

 is rather to be preferred, as more convenient for the purpose. Thus, if the 

 equation be by — 20/ + l6y^ = ^, it will be 3/ = -i- ^ ^\ + / — -rVinr 

 + 4- -^ "H— " V' — t-oVb"* And if by any means the 5th root of binomials can 

 be extracted, the root will come out true and possible, although the expression 

 seems to include an impossibility. Now the 5th root of the binomial -|4- + 

 y/ — -^-J-g\ is 4- ^ 4- 4/ — 15, ^nd the fifth root of the binomial -|4 — 

 y/ — ^Y^^g is ^ — ^ \/ ~ 15» and the half sum of these two roots is 4- "= y- 

 But if this extraction cannot be performed, or should seem too difficult, the 

 thing may always be effected by the table of natural sines in the following 

 manner. 



To the radius 1 let o = 44- = 0-95112 be the sign of a certain arc, which 

 therefore will be 72° 23'; the 5th part of which (because n = 5) is 14° 28'; 

 the sine of this is O.2498I = 4^ nearly. And the same way for equations of 

 higher degrees. 



Several Experiments showing the strange Effects of the Effluvia of Glass, pro- 

 ducible on its Motion and Attrition, By Mr. Fr. HauksbeCy F, R. S, 

 N" 309, p. 2372. 



ExpER. I. Containing further Observations on the Attrition of Glass. — 

 In the late experiments, which seem something to illustrate attraction or elec- 

 tricity, by the ends of the surrounding threads pointing to the axis of the 

 fricated glass, there is something further very remarkable ; viz. that after the 

 attrition of the glass has been a little while continued, and the effluvia has laid 

 hold on the hairy woolly threads, hanging loose, that then, though the rub- 

 bing was ceased, and the glass without motion, yet all the threads would con- 

 tinue their directed position for 4 or 5 minutes, and sometimes longer, before 

 they could disengage themselves from the attracting or electrical effluvia. And 

 if one's finger, or any thing else be approached near the pointing ends of the 

 threads, while the effluvia act with so much vigour, as to sustain them directed. 



