April 27, 1917] 



SCIENCE 



409 



give an anesthetic or a poison by way of the 

 mouth which is almost impossible. However, 

 substances can be introduced into the alimen- 

 tary tract through the anus and the desired 

 results obtained. 



Such is the method used in this laboratory. 

 Chloroform is injected into the cloaca and a 

 string tied in front of the anus to prevent the 

 ejection of the liquid. Five c.c. of chloroform 

 thus given will anesthetize an eight-inch 

 turtle sufficiently for dissection in thirty to 

 forty-five minutes. 



The value of this method is threefold. First, 

 a string and a pipette constitute the necessary 

 equipment; second, the ease with which the 

 anesthetic can be given is evident; and third, 

 there is no danger of the specimens coming 

 out from under the chloroform. 



ISTev^ton Miller 



XJniveesity op Utah 



SCIENTIFIC BOOKS 



C ombinatory Analysis. By Major Percy A. 



MaoMahon. Cambridge University Press. 



1915, Vol. 1, six -f 300 pp., and 1916, Vol. 2, 



is -f 340 pp. 



One of the four grand divisions of what 

 may be called properly static mathematics is 

 the theory of configurations. ■ It includes the 

 construction out of given elements of com- 

 pound forms under certain given conditions 

 or restrictions; together with the characters 

 possessed by such constructions when they are 

 varied under given laws, such as, for instance, 

 the character of transitivity, or that of primi- 

 tivity; and the laws of dependence of such 

 constructions upon each other; as well as 

 finally the invention of new or ideal elements 

 of mathematics that enable the solution of 

 problems of construction to be effected. These 

 constructions vary from the mere permutation 

 of a linear series of elements up to the com- 

 plicated trees of chemical combinations 

 studied by Cayley, and in general to all sorts 

 of problems in what has been happily denomi- 

 nated tactics by Cayley, or syntactics by 

 Cournot. "We find in its field the construction 

 of magic squares, of Latin squares, of Latin- 

 Greek squares, of triangles, stars, polygons. 



chess problems, routes over net works, prob- 

 lems of topography, and without much stretch 

 of imagination we might now include the dis- 

 position of the elements of war. The field is 

 obviously large in extent, and in a wide variety 

 of aspects fascinating. From certain points of 

 view one might be tempted to conclude that 

 we could include in it all mathematics, for the 

 definition given by C. S. Peirce made mathe- 

 matics the science of ideal constructions and 

 their applicability to the world as it is. 



The study of configurations usually begins 

 with combinatory analysis. By this is usually 

 meant the study of the arrangements along a 

 line of a collection of objects, either as indi- 

 viduals or in groups; arrangements at the 

 nodes of a lattice; combinations of arrange- 

 ments. Such problems arise not only as mat- 

 ters of tactic, curious problems or puzzles, but 

 in the determination of the nuraber of such 

 arrangements needed in solving problems in 

 the theory of probabilities. 



The treatise of Professor MacMahon under- 

 takes to present some very general methods 

 of handling such studies. These methods con- 

 sist for a large part in the construction of 

 enumerating generating functions, and in- 

 volve considerable study of symmetric func- 

 tions and certain differentiating operators. In 

 the course of this study he arrives at some 

 very elegant theorems. These methods not 

 only enumerate the possible forms, but in many 

 cases afford methods of actual construction 

 of the entire list of such possible forms. They 

 are very powerful and have enabled the author 

 to solve problems that were considered for a 

 long time to be beyond the reach of mathe- 

 matical analysis. His success and presenta- 

 tion in complete form may induce others to 

 study this important branch of mathematics. 



There are eleven sections, and the topics 

 under consideration will give some idea of 

 the character of the treatise. Section one con- 

 siders ordinary symmetric functions and their 

 connection with the theory of distribution of 

 objects into parcels. The operators which are 

 useful for these purposes are developed, and 

 their algebra considered, turning out to be 

 quite analogous to the algebra of symmetric 



