l62 



NATURE 



{Dec. 19, 1878 



(I).. 



quantities such that 



(11)^1 -f (I2)a-2+ ... + {ln)Xn = -?jV 

 (2i)jri +{22)x^-\- ... + (2n)xn ^ e.\ 



(SO-^l + (32) -^2 + •■• + (3'l)Xn = ^sV 

 («l)Xi + («2)X2+ ... + (««)x« = ^J 



We shall suppose JTi, jTj, ...jr„to be each so small that (11), 

 (12), ... (21), &c., do not vary sensibly from the values which they 

 have where Xj, x.,, ... jc,,, are each infinitely small. In practice 

 it will be convenient to so place the axes of B^, B^, . . . B„, and 

 the mountings of the pulleys on B^, B.^, ... B„, and the fixed 

 points Z?i, Eyi D.^, &c., that when x-^, Xn,...Xn are infinitely 

 small, the straight parts of each cord and the lines of infinitesi- 

 mal motion of the centres of the pulleys round which it passes are 

 all parallel. Then | (11), \ (21), ... \ («i) will be simply equal 

 to the distances of the centres of the pulleys P^^, P^^, . . . P„^, 

 from the axis of B^ ; 



i(i2), i (22) ... I («2) the distances of P^.^, P«^, ...P„^ from 

 the axis of B, and so on. 



In practice the mountings of the pulleys are to be adjustable 

 by proper geometrical slides, to allow any prescribed positive or 

 negative value to be given to each of the quantities (11), 

 (12), ... (21), &c. 



Suppose this to be done, and each of the bodies B-^^, B2, ... B„ 

 to be placed in its zero position and held there. Attach now 

 the cords firmly to the fixed points D^, D.^ . . . Z>„ respectively ; 

 and, passing them round their proper pulleys, bring them to the 

 other fixed points is"j, £^, ... E„, and pass them through infinitely 

 small smooth rings fixed at these points. Now hold the bodies 

 B^, B^y ... each fixed, and (in practice by weights hung on their 

 ends, outside -£"1, E^, ... E,i) piill the cords through -£"1, E^, ... En 

 with any given tensions^ T^, T^, 7'„. Let Gi, G^, ... Gn be 

 moments round the fixed axes of B-^, B2, ... B„ of the forces re- 

 quired to hold the bodies fixed when acted on by the cords thus 

 stretched. The principle of "virtual velocities," just as it came 

 from Lagrange (or the principle of " work"), gives immediately, 

 in virtue of (I), 



Ci = (II) 7\ + (21) 7; + ... + («i) 7;,^ 

 G.2 = (12) 71 + (22) 7\ + ... + («2) T„ 



.(II). 



Gn = (i«) T-, + (2«) Tg + ... + {nn) Tn 



Apply and l:eep applied to each of the bodies, B-^, B^, ... B„ 

 (in practice by the weights of the pulleys, and by counter-pulling 

 springs), such forces as shall have for their moments the values 

 Gi, G.2, ... G'„, calculated from equations (II) with whatever values 

 seem desirable for the tensions Ti, T„, ... Tn. (In practice, the 

 straight parts of the cords are to be approximately vertical, and the 

 bodies B^, B.^, ..., Bn^'i^ to be each balanced on its axis when the 

 pulleys belonging to it are removed, and it is advisable to make 

 the tensions each equal to half the weight of one of the pulleys 

 ■with its adjustable frame.) The machine is now ready for use. 

 To use it, pull the cords simultaneously or successively till 

 lengths equal to ^j, e,^, ... Cn are passed tlirough the rings E^, 

 E^, . . . En, respectively. 



The pulls required to do this may be positive or negative ; in 

 practice, they will be infinitesimal, downward or upward 

 pressm-es applied by hand to the stretching weights which 

 remain permanently hanging oh the cords. 



Observe the angles through which the bodies B^, B.^ . . . Bn 

 are turned by this given movement of the cords. These angles 

 are the required values of the unknown x-^, Xj, ...x„^ satis- 

 fying the simultaneous equations (i). 



The actual construction of a practically useful machine for 

 calculating as many as eight or ten or more of unknowns from 

 the same number of linear equations does not promise to be 

 either difficult or over-elaborate. A fair approximation being 

 found by a first application of the machine, a very moderate 

 amount of straightforward arithmetical work (aided very 

 advantageously by Crelle's multiplication tables) suffices to 

 calculate the residual errors, and allow the machines (with the 

 setting of the pulleys unchanged) to be re -applied to calculate the 

 corrections (which may be treated decimally, for convenience) : 

 thus, 100 times the amount of the correction on each of the 



« The idea of force here first introduced is not essential, indeed is not tech- 

 nically admissible to the purely kinematic and algebraic part of the subject 

 proposed. But it is not merely an ideal kinematic construction of the alge- 

 braic problem that is intended ; and the design of a kinematic machine, for 

 success in practice, essentially involves dynamical considerations. In the 

 present case some of the most important of the purely algebraic questions 

 concerned are very interestingly illustrated by these dynaniical considerations. 



original unknowns, to be made the new unknowns, if the magni- 

 tudes thus falling to be dealt with are convenient for the machme. 

 There is, of course, no limit to the accuracy thus obtainable by 

 successive approximations. The exceeding easiness of each 

 application of the machine promises well for its real usefulness, 

 whether for cases in which a single application suffices, or for 

 others in which the requisite accuracy is reached after two, three, 

 or more of successive approximations. 



Mathematical Society, December 12.— Mr. C, W. Merri- 

 field, F.R.S., president, in the chair.— Prof. W. S. Jevons, 

 F.R.S,, was elected a Member. — The following communica- 

 tions were made to the Society : — Mr. H. Perigol, on a kine- 

 matic paradox (the rotameter); Mr. S. Roberts, F.R.S. , on 

 the forms of numbers determined by continued fractions ; Prince 

 Camille de Polignac, on a graphic construction of the powers of 

 a linear substitution. 



Linnean Society, December 5.— Prof. AUman, F.R.S., 

 president, in the chair. — Dr. I. Bayley Balfour demonstrated 

 the peculiarities of a rare Myxomycetes, which species of Hete- 

 rodictyum he showed bore characters intermediate between 

 Cribraria and Dictydium. — Mr. G. Murray called attention to a 

 peculiar greeni=h-yellow fungus {Hygrophorus Wynnicc, Berk. ?) 

 from Bridlington, Yorkshire. — Examples of a moss new to 

 Britain, the Aulacomnion turgidjim, were shown by Mr. E. M. 

 Holmes, who stated that they were found by Mr. West and Dr. 

 F. Arnold Lees in Yorkshire ; a comparison between the above 

 and the common A. fahistre was made. — Mr. F. H. Water- 

 house read a paper on some Coleoptera collected by Charles 

 Darwin, of geographical interest. These had lain undetermined 

 for a long series of years, and now prove new to science. 

 Phytosiis Danuinii, from the Falklands, has unusually long, 

 slender claws ; Choleva falklandica is elliptical-shaped and 

 strongly punctated. Elmts brunnea and Ant/iicus Wollastoni, 

 from St. Helena, are noteworthy, inasmuch as Mr. Wollaston 

 (" Coleop. St. Hel.") does not record either genus as existent 

 there. Scaphisoma elongatum, from Rio de Janeiro, is the 

 first species of the genus known to inhabit South America ; 

 and Prosthetops (P. capensis) is a novel genus with two ocelli, 

 from South Africa. — Mr. C. B. Clarke, in a note on Gardeuia 

 iurgida, stated that in books the flower calyx of males was alone 

 described, while all herbaria specimens are dioecious, and males 

 and females have hitherto been referred to different genera. 

 The precise characters of each were denoted. — Dr. F. Day gave 

 a summary of his (third) concluding paper on the geographical 

 distribution of the Indian fresh-water fishes, in this dealing 

 with the families Scombresocidse, Cypriodontida;, Cyprinidre, 

 Notopteridoe, and Symbranchidee, Among the eighty-seven 

 genera two only are African, thirty-two extend to the Malay 

 Archipelago, and twelve are common to Africa and IMalaya ; of 

 369 species two are African, twenty-seven Malayan, and two 

 common to both regions. In short, the fresh-water fith affinities 

 preponderate to the Indo-Chinese and Malayan sub-regions ; thus 

 supporting Mr, Wallace's opinion as opposed to the view held 

 by Mr, Blandford, who gives greater weight to African relation- 

 ships, at least so far as mammals are concerned, Dr, Day, 

 moreover, contends that the Indian fresh-water fishes point to 

 three subordinate separate faunas — i. That belonging to the 

 Ghauts, Ceylon, the Himalayas, and Malay Archipelago ; 

 wherein may be distinguished two fish races, a Palrearctic and a 

 Malayan. 2. A fauna of the plains west of the Indus, with an 

 African element in it. 3. That (by far the largest) spread over 

 the plains east of the Indus, and which appears to have a Bur- 

 mese connection. — The abstract was read of a second contribu- 

 tion on the mollusca of the Challenger Expedition, by the Rev. 

 R. Boog Watson. This consisted of descriptions of species of 

 Trochidse belonging to four genera, viz., Sequinzia, Basilissa, 

 Gaza, and Bembix ; the three last being new and otherwise re- 

 markable. — Messrs. Dowdeswell, Arthur Hammond, Thos. 

 Hanbury, Joseph Sidebotham, Wm. Thomson, and Chas. A, 

 Wright were elected Fellows of the Society. 



Zoological Society, December 3. — Mr. Robert Hudson> 

 F.R.S., vice-president, in the chair. — Mr. H. Seebohm, F.Z.S., 

 exhibited a series of specimens of the hooded and carrion crows, 

 and made remarks on their intermediate forms and geographical 

 distribution. — Col, L, H. Loyd Irby, F.Z.S,, exhibited and 

 made remarks on the nests, eggs, and young of Cypselus palli- 

 dus, taken at Gibraltar, — Mr, Howard Saunders, F.Z,S.,i 

 exhibited and made remarks on some eggs of Indian Laridael 

 {Sterna bergii and Larus hemprichii), which had been taken byj 



