20O CLASS OF altf:rnants. 



From these there drop into our hands, as it were, the results of 

 the evaluation of fifteen alternants of the sixth order. 



(7) The corresponding investigation in the case where the number 

 of angles is odd is somewhat more complicated, and has to be at 

 full}' entered into as if the other case had not been touched. Is 

 must suffice to give merely the main results in two cases. 



When there are three angles the fundamental equalitv is 



A + F ri,s 2o + L SI 71 2(t = —2 rot a . IT s//? a 

 where A = ros (a+ jS + y) — <os u ros /^ cos y, 



L = .s//? (a 4-/^ + 7) : 



and the determinant of the coefficients of A, F, L. 



I 1 roa 2(1 sin 2o | =2^ sin (y— /3) sin (y — a) sin (^_f,). 



When there are fire angles the fundamental equality is 



A + F ros2a + G rosAu + L sin 2o + M sinAa = — 8 rot a . II sin a 



(XVIII) 

 where A=S <'os (So — 2a) — 8 II ros «, 



F = S ''f^s (^ct—2a) — ros 2«, G= — ros So, 



L=S ■'^'iii (So — 2a) — '<in S«' M= — sin So; 



tlie related auxiliary equality is 



16 II ros a=ros Sa + Sros (So-2o) + S '"^.s (So-2o-2/3) (XIX) ; 



and the determinant of the coefficients of A, F, G, L, M 



I 1 r(js 2h ros 4o s/n 2a SI/I in \ 

 ==-2\Us n(t-o) (XX). 



It will be observed that in both these cases there is no longer 

 any difficulty (see § 4) in finding the alternant expressions for tlie 

 sine and cosine of the sum of the angles, these functions now 

 occurring themselves among the so-called " unknowns " 



Fossil Man in Florida. — 'J'hc Amcncan Jonmal of 



Science ( 1916), 1-18, jjublishes a report on fi)ssil human remain.s 

 recently discovered at Vero. on the Atlantic coast of central- 

 eastern i'lorida. 1 lie results were found associated with verte- 

 b.ate, fresh-water invertebrate, and plant fossils. In all twelve 

 s]>L'cies of vertebrate remains were found, including;- the fox. deer, 

 sloth, and mastodon. In the older stream de])Osits fossils 

 characteristic of the .\ortli American IMeistocene have been 

 found, anioui^st which \\(,tc vcrtebrji' and teeth of a lari^c and 

 probably extinct crocodile. It is assumed that the human bones 

 were contemporaneous with the fauna re])resented, and it is held 

 that the bed in which the\- were found was nn(!r)nbtcdh- de])osited 

 durinif the Pleistocene. 



