AMERICAN INSTITUTE. 505 



dion, and turning the flask round to have all parts covered equally with it. 

 It dries soon, and can be drawn out and filbd with hydrogen. Such bal- 

 loons look beautiful, and last a few minutes in the air. 



The discussion was carried on for some time longer in the form of a con- 

 versation, after which the subject for the next evening was selected, viz. ; 

 "Head coverings, and the machinery employed in their manufacture." 



The Association then adjourned. 



PoLYTEcnNic Association of the American Institute, 



December 8, 1859. 



President Mason in the chair. Mr. Garvey, Secretary, pro tem. 



The Miscellaneous business being called up, 



Mr. Fisher read a short communication from himself to the club, on 

 steam-wagons on common roads, with especial reference to the employment 

 of steam omnibuses in cities. He detailed his own efforts to obtain per- 

 mission from the city authorities of New York to run a steam hackney 

 coach in this city, and the legal quibbles by which he was opposed, after 

 which he proposed to the Polytechnic Club to use their influence as a 

 scientific body to bring about the formation of a club, who should be joint 

 stock owners of one or more wagons, which they and their friends could 

 use as private vehicles, unless prosecuted as nuisances, in which case evi- 

 dence should be given to prove them such. 



Mr. Fisher's views were warmly supported by the meeting, and Messrs. 

 Seeley, Johnson, and Butler, were appointed a committee to aid in carry- 

 ing them out. 



Prof. Hedrick took up some properties of the natural numbers which Mr. 

 Tillman had introduced to the club on a former occasion, and gave the fol- 

 lowing demonstration of the proposition that " The square of the sum of 

 the natural numbers, up to any point, is equal to the sum of the cubes of the 

 natural numbers taken up to that point." 



The demonstration was as follows, viz : — 



Let S = the sum of the natural numbers to ?z, 

 and S'= •' " ion-\-\ 



then S :^n-{- In (7i-~l) 



S' — S -= 1 + «, 



S' + S = (1 + n)„ 

 (S'_S) (S' + S) = S,-S, = (l + n\, 



S', = S, + (1 + «)„ 

 That is having the square of the sum of the natural numbers up to any 

 point, as n, by adding the cube of the next number, we obtain the sum up 

 to the next higher number, or to ?z -|- 1. 



If« = l, thenS = l, and S' = 3, 

 M + 1 = 2, 

 S', = 3, = 1, + (;. + 1)3 =. 1 4- 8 = 9 = 1, -h 2„ 

 Therefore we have the general equation 



(1 + 2 + 3 + &c. + 7i\ = 1, + 2, + 3, 4- &c. 4- n,. 



