July 29, 1898.] 



SCIENCE. 



121 



(1 ) White Sands, 9 ears, weigbiiig 10.38 grams, containing 283 grains, weighing 7.310 grms. 



(2) Larrea, 10 " " 9.77 " " 243 " " 6.995 " 



(3) Mesquite, 10 " " 10.60 " " 285 " • " 7.610 " 



(4) Tornillo, 10 " " 11.88 " " 330 " " 8.270 " 



(5) Adobe, 10 " " 10.24 " " 254 " ' ' 7.335 " 



Peas (Measurements in cm.). 



No. 1 and 4 were not perfectly ripe, and 

 may owe a very little of their weight to the 

 extra moisture they contain. It will be 

 seen that the gypsum wheat weighs up well 

 with the others, and when its green heads, 

 above mentioned, are ripe the product will 

 outweigh considerably all of the others. 



It is seen from the table that the gypsum 

 peas are decidedly the best. "We could not 

 measure the yield (the gypsum peas were 

 the first to bloom, on March 19th) , because 

 certain girls of the class in horticulture saw 

 fit to remove some of the pods when unob- 

 served by their professor. 



Conclusions. — It appears, from these pre- 

 liminary researches, that nearly pure gyp- 

 sum will nourish plants as well as ordinary 

 soil, or even better. It is not apparent how 

 the wheat, etc., come by their nitrogen in 

 such a soil, though the peas may well get it 

 by means of their root-tubercles. The ab- 

 sence of other elements is also noticeable, 

 but it is not worth while at the present 

 time to enter into a detailed discussion of 

 causes and effects, as further researches will, 

 it is hoped, make such a discussion more 

 profitable at a later date. 



t. d. a. cockeeell. 

 Fabian Garcia. 



N. M. Age. Esp. Sta. 



Mesilla Park, N. M., June 17, 1898. 



THE CROSS-RATIO GROUP OF 120 QUADRATIC 



CREMONA TRANSFORMATIONS OF 



THE PLANE* 



Geoups of linear substitutions have long 

 been studied with reference to (1) the geo- 

 metric representation in the plane or on the 

 sphere, (2) the rational integral functions 

 left invariant under the operators of the 

 group. These questions now prove to be of 

 interest when investigated for groups of 

 transformations of order higher than the 

 first. The theory of birational transforma- 

 tions (quadratic and higher) has been given 

 by Cremona, Cayley, Clebsch and others. 

 Groups of such transformations have been 

 enumerated by Autonne and S. Kantor. 

 The cross-ratio Cremona transformation 

 groups of order n ! were first given by Pro- 

 fessor E. H. Moore in his lectures at the 

 University of Chicago in the spring of 1895. 

 These groups are found by determining foi 

 each permutation of n quantities a funda- 

 mental system of n-3 cross-ratios in terms 

 of which the cross-ratios of every four out of 

 the n quantities are expressible, and then 

 setting up the transformation relations 

 among these n ! fundamental systems. 



* Abstract of a Dissertation submitted to the Facul- 

 ties of the Graduate Schools of Arts, Literature and 

 Science in the University of Chicago, April, 1898, in 

 candidacy tor the degree of Doctor of Philosophy 

 (Department of Mathematics), by H. E. Slaught. 



