OF ARTS AND SCIENCES. 241 



of a bucket of a turbine wheel, constructed by Mr. U. A. 

 Boyden, experiments upon which, conducted with the most 

 scrupulous care, had shown it to produce an efiect equal to 

 eighty-eight per cent, of the power expended ; and stated that 

 some of Mr. Boyden's wheels had given the astonishing result 

 of ninety-two per cent, of the power. 



Professor Peirce made some remarks in regard to the frac- 

 tion which expresses the law of vegetable growth, which he 

 compared with the ratio of the mean motions of the planets, 

 and found to express more nearly the arrangement of these 

 bodies than Bode's law. For this purpose, Neptune's period 

 of revoUition must be multiplied by y X i to obtain that of 

 Uranus. The period of Uranus must be multiplied by -} X 1 

 to obtain that of Saturn. Saturn's period must be multiplied 

 by f X f to obtain that of Jupiter, and so on. If this law is 

 true, there can be only one planet within the orbit of Mercury, 

 and no planet beyond Neptune. This law or harmony seems 

 to be that to which successive development in general tends 

 to conform, and is manifested when other forces opposed to it 

 are not too powerful. The atomic laws are opposed to it, in 

 crystallizing and other chemical processes ; and also the higher 

 laws of organization, such as those of bilateral division in the 

 higher animals. 



Professor Peirce remarked that the perturbative function of 

 planetary motion had been developed by Hansen, according 

 to the eccentric anomaly of one of the planets, in a numerical 

 form ; and exhibited the first terms of a literal development of 

 this function, which is more simple than the usual form of 

 development according to the mean anomaly. He thought 

 there were reasons for believing that some other form of de- 

 velopment will be discovered better adapted to cases of great 

 inclinations and eccentricities ; inasmuch as, in case the two 

 orbits do not approach each other within a small distance, the 

 development of this function should not contain any term 

 capable of becoming infinite. 



Dr. C. T. Jackson laid before the Academy a number of 



VOL. II. 31 



