The literature on Balanchine is vast, but it is primarily biographical. Balanchine Variations is the first book to concentrate on the ballets themselves, providing critical analysis and detailed descriptions of what the dancers actually do.   Beginning with Apollo (1928), Balanchine's first extant work, and ending with one of his last ballets, Ballo della Regina (1978), Nancy Goldner offers detailed insights into more than twenty individual ballets. Based on lectures given across the United States, under the auspices of the Balanchine Foundation, they are intended to illuminate his art.   Goldner discusses the history of each ballet, places each in the context of Balanchine's life and sensibility. She also addresses his taste in music and whether his style can be considered particularly American.   The ballets Balanchine choreographed for the New York City Ballet are danced by companies around the world, and this innovative book is sure to become an indispensable guide to dancers and spectators alike.